numerical studies on quantized vortex dynamics in ... · quantized vortices, which are the...
TRANSCRIPT
NUMERICAL STUDIES ON QUANTIZED
VORTEX DYNAMICS IN SUPERFLUIDITY
AND SUPERCONDUCTIVITY
TANG QINGLIN
NATIONAL UNIVERSITY OF SINGAPORE
2013
NUMERICAL STUDIES ON QUANTIZED
VORTEX DYNAMICS IN SUPERFLUIDITY
AND SUPERCONDUCTIVITY
TANG QINGLIN
(B.Sc., Beijing Normal University)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF MATHEMATICS
NATIONAL UNIVERSITY OF SINGAPORE
2013
Acknowledgements
It is my great honour to take this opportunity to thank those who made this thesis
possible.
First and foremost, I owe my deepest gratitude to my supervisor Prof. Bao Weizhu,
whose generous support, patient guidance, constructive suggestion, invaluable help and
encouragement enabled me to conduct such an interesting research project.
I’d like to express my appreciation to my collaborators Asst. Prof. Zhang Yanzhi, Dr.
Daniel Marahrens and Dr. Jiang Wei for their contribution to the work. Specially, I thank
Dr. Zhang Yong for reading drafts. My sincere thanks go to all the former colleagues and
fellow graduates in our group, especially Dr. Dong Xuanchun and Prof. Wang Hanquan
for fruitful discussions and suggestions on my research. I heartfeltly thank my friends,
especially Zeng Zhi, Xu Weibiao, Feng Ling, Yang Lina, Qin Chu, Zhu Guimei and Wu
Miyin, for all the encouragement, emotional support, comradeship and entertainment they
offered. I would also like to thank NUS for awarding me the Research Scholarship for the
financial support during my Ph.D candidature. Many thanks go to IPAM at UCLA and
WPI at University of Vienna for their financial assistance during my visits.
Last but not least, I am forever indebted to my beloved wife and family, for their
encouragement, steadfast support and endless love when it was most needed.
Tang Qinglin
March 2013
i
Contents
Acknowledgements i
Summary iv
List of Tables vii
List of Figures viii
List of Symbols and Abbreviations xii
1 Introduction 1
1.1 Vortex in superfluidity and superconductivity . . . . . . . . . . . . . 1
1.2 Problems and contemporary studies . . . . . . . . . . . . . . . . . . . 3
1.2.1 Ginzburg-Landau-Schrodinger equation . . . . . . . . . . . . . 3
1.2.2 Gross-Pitaevskii equation with angular momentum . . . . . . 9
1.3 Purpose and scope of this thesis . . . . . . . . . . . . . . . . . . . . . 12
2 Methods for GLSE on bounded domain 14
2.1 Stationary vortex states . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Reduced dynamical laws . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Under homogeneous potential . . . . . . . . . . . . . . . . . . 17
ii
Contents iii
2.2.2 Under inhomogeneous potential . . . . . . . . . . . . . . . . . 22
2.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Time-splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Discretization in a rectangular domain . . . . . . . . . . . . . 25
2.3.3 Discretization in a disk domain . . . . . . . . . . . . . . . . . 28
3 Vortex dynamics in GLE 31
3.1 Initial setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Numerical results under Dirichlet BC . . . . . . . . . . . . . . . . . . 33
3.2.1 Single vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.2 Vortex pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.3 Vortex dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.4 Vortex cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.5 Steady state patterns of vortex clusters . . . . . . . . . . . . . 42
3.2.6 Validity of RDL under small perturbation . . . . . . . . . . . 46
3.3 Numerical results under Neumann BC . . . . . . . . . . . . . . . . . 47
3.3.1 Single vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.2 Vortex pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.3 Vortex dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.4 Vortex cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.5 Steady state patterns of vortex clusters . . . . . . . . . . . . . 55
3.3.6 Validity of RDL under small perturbation . . . . . . . . . . . 56
3.4 Vortex dynamics in inhomogeneous potential . . . . . . . . . . . . . 57
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Vortex dynamics in NLSE 61
4.1 Numerical results under Dirichlet BC . . . . . . . . . . . . . . . . . 61
4.1.1 Single vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.2 Vortex pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.3 Vortex dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Contents iv
4.1.4 Vortex cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1.5 Radiation and sound wave . . . . . . . . . . . . . . . . . . . . 75
4.2 Numerical results under Neumann BC . . . . . . . . . . . . . . . . . 78
4.2.1 Single vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.2 Vortex pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.3 Vortex dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.4 Vortex cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.5 Radiation and sound wave . . . . . . . . . . . . . . . . . . . . 85
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Vortex dynamics in CGLE 88
5.1 Numerical results under Dirichlet BC . . . . . . . . . . . . . . . . . . 89
5.1.1 Single vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1.2 Vortex pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.3 Vortex dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1.4 Vortex cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.5 Steady state patterns of vortex clusters . . . . . . . . . . . . . 97
5.1.6 Validity of RDL under small perturbation . . . . . . . . . . . 101
5.2 Numerical results under Neumann BC . . . . . . . . . . . . . . . . . 102
5.2.1 Single vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.2 Vortex pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2.3 Vortex dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.4 Vortex cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.5 Validity of RDL under small perturbation . . . . . . . . . . . 110
5.3 Vortex dynamics in inhomogeneous potential . . . . . . . . . . . . . 112
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6 Numerical methods for GPE with angular momentum 116
6.1 GPE with angular momentum . . . . . . . . . . . . . . . . . . . . . . 116
6.2 Dynamical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Contents v
6.2.1 Conservation of mass and energy . . . . . . . . . . . . . . . . 118
6.2.2 Conservation of angular momentum expectation . . . . . . . . 119
6.2.3 Dynamics of condensate width . . . . . . . . . . . . . . . . . . 121
6.2.4 Dynamics of center of mass . . . . . . . . . . . . . . . . . . . 124
6.2.5 An analytical solution under special initial data . . . . . . . . 125
6.3 GPE under a rotating Lagrangian coordinate . . . . . . . . . . . . . . 126
6.3.1 A rotating Lagrangian coordinate transformation . . . . . . . 126
6.3.2 Dynamical quantities . . . . . . . . . . . . . . . . . . . . . . . 128
6.4 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4.1 Time-splitting method . . . . . . . . . . . . . . . . . . . . . . 133
6.4.2 Computation of Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . 135
6.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.5.1 Comparisons of different methods . . . . . . . . . . . . . . . . 140
6.5.2 Dynamics of center of mass . . . . . . . . . . . . . . . . . . . 142
6.5.3 Dynamics of angular momentum expectation and condensate
widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.5.4 Dynamics of quantized vortex lattices . . . . . . . . . . . . . . 148
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7 Conclusion remarks and future work 151
Bibliography 156
List of Publications 173
Summary
Quantized vortices, which are the topological defects that arise from the order
parameters of the superfluid, superconductors and Bose–Einstein condensate (BEC),
have a long history that begins with the study of liquid Helium. Their appearance
is regarded as the key signature of superfluidity and superconductivity, and most of
their phenomenological properties have been well captured by the Ginzburg-Landau-
Schrodinger equation (GLSE) and the Gross-Pitaevskii equation (GPE).
The purpose of this thesis is twofold. The first is to conduct extensive numerical
studies for the vortex dynamics and interactions in superfluidity and superconduc-
tivity via solving GLSE on different bounded domains in R2 and under different
boundary conditions. The second is to study GPE both analytically and numeri-
cally in the whole space.
This thesis mainly contains two parts. The first part is to investigate vortex
dynamics and their interaction in GLSE on bounded domain. We begin with the
stationary vortex state of the GLSE, and review various reduced dynamical laws
(RDLs) that govern the motion of the vortex centers under different boundary con-
ditions and prove their equivalence. Then, we propose accurate and efficient numer-
ical methods for computing the GLSE as well as the corresponding RDLs in a disk
vi
Summary vii
or rectangular domain under Dirichlet or homogeneous Neumann boundary condi-
tion (BC). These methods are then applied to study the various issues about the
quantized vortex phenomena, including validity of RDLs, vortex interaction, sound-
vortex interaction, radiation and pinning effect introduced by the inhomogeneities.
Based on extensive numerical results, we find that any of the following factors: the
value of ε, the boundary condition, the geometry of the domain, the initial location
of the vortices and the type of the potential, affect the motion of the vortices sig-
nificantly. Moreover, there exist some regimes such that the RDLs failed to predict
correct vortex dynamics. The RDLs cannot describe the radiation and sound-vortex
interaction in the NLSE dynamics, which can be studied by our direct simulation.
Furthermore, we find that for GLE and CGLE with inhomogeneous potential, vor-
tices generally move toward the critical points of the external potential, and finally
stay steady near those points. This phenomena illustrate clearly the pinning effect.
Some other conclusive experimental findings are also obtained and reported, and
discussions are made to further understand the vortex dynamics and interactions.
The second part is concerned with the dynamics of GPE with angular momentum
rotation term and/or the long-range dipole-dipole interaction. Firstly, we review
the two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduc-
tion under anisotropic external potential and derive some dynamical laws related
to the 2D and 3D GPE. By introducing a rotating Lagrangian coordinate system,
the original GPEs are re-formulated to the GPEs without the angular momentum
rotation. We then cast the conserved quantities and dynamical laws in the new
rotating Lagrangian coordinates. Based on the new formulation of the GPE for
rotating BECs in the rotating Lagrangian coordinates, we propose a time-splitting
spectral method for computing the dynamics of rotating BECs. The new numerical
method is explicit, simple to implement, unconditionally stable and very efficient in
computation. It is of spectral order accuracy in spatial direction and second-order
accuracy in temporal direction, and conserves the mass in the discrete level. Ex-
tensive numerical results are reported to demonstrate the efficiency and accuracy
Summary viii
of the new numerical method. Finally, the numerical method is applied to test the
dynamical laws of rotating BECs such as the dynamics of condensate width, angular
momentum expectation and center-of-mass, and to investigate numerically the dy-
namics and interaction of quantized vortex lattices in rotating BECs without/with
the long-range dipole-dipole interaction.
List of Tables
6.1 Spatial discretization errors ‖ψ(t)− ψ(hx,hy,∆t)(t)‖l2 at time t = 1.5. . 141
6.2 Temporal discretization errors ‖ψ(t)− ψ(hx,hy,∆t)(t)‖l2 at time t = 1.5. 142
ix
List of Figures
2.1 Plot of the function f εm(r) in (2.4) with R0 = 0.5. left: ε = 140
with
different winding number m. right: m = 1 with different different ε. . 15
2.2 Surf plot of the density |φεm|2 (left column) and the contour plot of
the corresponding phase (right column) for m = 1 (a) and m = 4 (b). 16
3.1 (a)-(b): Trajectory of the vortex center in GLE under Dirichlet BC
when ε = 132
for cases I-VI (from left to right and then from top to
bottom), and (c): dε1 for different ε for cases II, IV and VI (from left
to right) in section 3.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Contour plots of |ψε(x, t)| at different times for the interaction of
vortex pair in GLE under Dirichlet BC with ε = 132
and different
h(x) in (2.6): (a) h(x) = 0, (b) h(x) = x+ y. . . . . . . . . . . . . . 36
3.3 Trajectory of vortex centers (left) and time evolution of the GL func-
tionals (right) for the interaction of vortex pair in GLE under Dirich-
let BC with ε = 132
for different h(x) in (2.6): (a) h(x) = 0, (b)
h(x) = x+ y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
x
List of Figures xi
3.4 Time evolution of xε1(t) and xr1(t) (left and middle) and their dif-
ference dε1 (right) for different ε for the interaction of vortex pair in
GLE under Dirichlet BC for different h(x) in (2.6): (a) h(x) = 0, (b)
h(x) = x+ y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Contour plots of |ψε(x, t)| at different times for the interaction of
vortex dipole in GLE under Dirichlet BC with ε = 132
for different d0
and h(x) in (2.6): (a) h(x) = 0, d0 = 0.5, (b) h(x) = x+ y, d0 = 0.5,
(c) h(x) = x+ y, d0 = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 (a)-(c): Trajectory of vortex centers (left) and time evolution of the
GL functionals (right) for the interaction of vortex dipole in GLE
under Dirichlet BC with ε = 132
for different d0 and h(x) in (2.6): (a)
h(x) = 0, d0 = 0.5, (b) h(x) = x + y, d0 = 0.5, (c) h(x) = x + y,
d0 = 0.3. (d): Critical value dεc for different ε when h(x) ≡ x+ y. . . 39
3.7 Time evolution of xε1(t), xr1(t) (left and middle) and their difference dε1
(right) for different ε for the interaction of vortex dipole in GLE under
Dirichlet BC with d0 = 0.5 for different h(x) in (2.6): (a) h(x) = 0,
(b) h(x) = x+ y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.8 Critical value dεc for the interaction of vortex dipole of the GLE under
Dirichlet BC with h(x) ≡ x+ y in (2.6) for different ε. . . . . . . . . 40
3.9 Trajectory of vortex centers for the interaction of different vortex
clusters in GLE under Dirichlet BC with ε = 132
and h(x) = 0 for
cases I-IX (from left to right and then from top to bottom) in section
3.2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.10 Contour plots of |φε(x)| for the steady states of vortex cluster in
GLE under Dirichlet BC with ε = 116
for M = 8, 12, 16, 20 (from
left column to right column) and different domains: (a) unit disk
D = B1(0), (b) square domain D = [−1, 1]2, (c) rectangular domain
D = [−1.6, 1.6]× [−1, 1]. . . . . . . . . . . . . . . . . . . . . . . . . . 43
List of Figures xii
3.11 Contour plots of |φε(x)| for the steady states of vortex cluster in
GLE under Dirichlet BC with ε = 116
on a rectangular domain D =
[−1.6, 1.6]× [−1, 1] for M = 8, 12, 16, 20 (from left column to right
column) and different h(x): (a) h(x) = 0, (b) h(x) = x + y, (c)
h(x) = x2 − y2, (d) h(x) = x− y, (e) h(x) = x2 − y2 − 2xy. . . . . . . 44
3.12 Contour plots of |φε(x)| for the steady states of vortex cluster in GLE
under Dirichlet BC with ε = 116
and M = 8 on a unit disk D = B1(0)
(top row) or a square D = [−1, 1]2 (bottom row) under different
h(x) = 0, x + y, x2 − y2, x− y, x2 − y2 − 2xy (from left column to
right column). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.13 Width of the boundary layer LW vsM (the number of vortices) under
Dirichlet BC on a square D = [−1, 1]2 when ε = 116
for different h(x):
(a) h(x) = 0, (b) h(x) = x+ y. . . . . . . . . . . . . . . . . . . . . . 45
3.14 Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 3.2.6 . . . . . . . . . . . . . . . . 47
3.15 Trajectory of the vortex center when ε = 132
(left column) and dε1
for different ε (right column) for the motion of a single vortex in
GLE under homogeneous Neumann BC with different x01 in (2.6): (a)
x01 = (0, 0.1), (b) x0
1 = (0.1, 0.1). . . . . . . . . . . . . . . . . . . . . . 48
3.16 Dynamics and interaction of a vortex pair in GLE under Neumann
BC: (a) contour plots of |ψε(x, t)| with ε = 132
at different times, (b)
trajectory of the vortex centers (left) and time evolution of the GL
functionals (right) for ε = 132, (c) time evolution of xε1(t) and xr1(t)
(left and middle) and their difference dε1(t) (right) for different ε. . . 50
3.17 Contour plots of |ψε(x, t)| at different times for the interaction of
vortex dipole in GLE under Neumann BC with ε = 132
for different
d0: (a) d0 = 0.2, (b) d0 = 0.7. . . . . . . . . . . . . . . . . . . . . . . 51
List of Figures xiii
3.18 Trajectory of vortex centers (left) and time evolution of the GL func-
tionals (right) for the interaction of vortex dipole in GLE under Neu-
mann BC with ε = 132
for different d0: (a) d0 = 0.2, (b) d0 = 0.7. . . . 51
3.19 Time evolution of xε1(t) and xr1(t) (left and middle) and their differ-
ence dε1(t) (right) for different ε and d0: (a) d0 = 0.2, (b) d0 = 0.7. . . 52
3.20 Trajectory of vortex centers for the interaction of different vortex
clusters in GLE under homogeneous Neumann BC with ε = 132
for
cases I-IX (from left to right and then from top to bottom) in section
3.3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.21 Contour plots of the amplitude |ψε(x, t)| for the initial data (top) and
corresponding steady states (bottom) of vortex cluster in the GLE
under homogeneous Neumann BC with ε = 116
for different number
of vorticesM and winding number nj : M = 3, n1 = n2 = n3 = 1 (first
and second columns); M = 3, n1 = −n2 = n3 = 1 (third column);
and M = 4, n1 = −n2 = n3 = −n4 = 1 (fourth column). . . . . . . . 55
3.22 Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 3.3.6 . . . . . . . . . . . . . . . . 57
3.23 (a) and (b): trajectory, time evolution of the distance between the
vortex center and potential center and dε1(t) for different ε for case
I and II, and (c): Trajectory of vortex center for different ε of the
vortices for case III in section 3.4. . . . . . . . . . . . . . . . . . . . 59
4.1 Trajectory of the vortex center in NLSE under Dirichlet BC when
ε = 140
for Cases I-VI (from left to right and then from top to bottom
in top two rows), and dε1 for different ε for Cases I,V&VI (from left
to right in bottom row) in section 4.1.1. . . . . . . . . . . . . . . . . . 64
4.2 Trajectory of the vortex center in NLSE dynamics under Dirichlet
BC when ε = 164
(blue solid line) and from the reduced dynamical
laws (red dash line) for Cases VI-XI (from left to right and then from
top to bottom) in section 4.1.1. . . . . . . . . . . . . . . . . . . . . . 65
List of Figures xiv
4.3 Trajectory of the vortex center in NLSE under Dirichlet BC when
ε = 140
for cases I, XII-XIII, VI and XIV-XV (from left to right and
then from top to bottom) in section 4.1.1. . . . . . . . . . . . . . . . 66
4.4 Form left to right in (a)-(c): trajectory of the vortex pair, time evo-
lution of Eε(t) and Eεkin(t) as well as xε1(t) and xε2(t) for the 3 cases
in section 4.1.2. (a). case I, (b). case II, (c). case III. (d). time
evolution of dε1(t) for case I-III (form left to right). . . . . . . . . . . . 67
4.5 Critical value dεc for the interaction of vortex pair of the NLSE under
the Dirichlet BC with different ε and h(x) = 0 in (2.6): if d0 < dεc,
the two vortex will move along a circle-like trajectory, if d > dεc, the
two vortex will move along a crescent-like trajectory. . . . . . . . . . 68
4.6 Contour plots of |ψε(x, t)| at different times (top two rows) as well as
the trajectory, time evolution of xε1(t), xε2(t) and dε1(t) (bottom two
rows) for the dynamics of a vortex dipole with different h(x)in section
4.1.3: (1). h(x) = 0 (top three rows), (2). h(x) = x+ y (bottom row). 69
4.7 Trajectory of the vortex xε1 (blue line), xε2 (dark dash-dot line) and
xε3 (red dash line) (first and third rows) and their corresponding time
evolution (second and fourth rows) for Case I (top two rows) and
Case II (bottom two rows) for small time (left column), intermediate
time (middle column) and large time (right column) with ε = 140
and
d0 = 0.25 in section 4.1.4. . . . . . . . . . . . . . . . . . . . . . . . . 70
4.8 Contour plots of |ψε(x, t)| with ε = 116
at different times for the
NLSE dynamics of a vortex cluster in Case III with different initial
locations: d1 = d2 = 0.25 (top two rows); d1 = 0.55, d2 = 0.25
(middle two rows); d1 = 0.25, d2 = 0.55 (bottom two rows) in section
4.1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
List of Figures xv
4.9 Contour plots of −|ψε(x, t)| ((a) & (c)) and the corresponding phase
Sε(x, t) ((b) & (d)) as well as slice plots of |ψε(x, 0, t)| ((e) & (f)) at
different times for showing sound wave propagation under the NLSE
dynamics of a vortex cluster in Case IV with d0 = 0.5 and ε = 18in
section 4.1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.10 Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 4.1.5 . . . . . . . . . . . . . . . . 73
4.11 Trajectory of the vortex xε1(t) (blue dash-line), xε2(t) (red solid line)
and xε3(t) (dark dash-dot line) (top row) for δ = 0.0005 as well as
time evolution of the distance of xε2 to origin, i.e. |xε2(t)|, (bottomrow) for Type I (first column), Type II (second column) and Type III
(third column) perturbation on the initial location of the vortices for
different δ in section 4.1.4, where ε = 140
and d0 = 0.25. . . . . . . . 75
4.12 Surface plots of −|ψε(x, t)| ((a) & (c)) and contour plots of the corre-
sponding phase Sε(x, t) ((b) & (d)) as well as slice plots of |ψε(x, 0, t)|((e) & (f)) at different times for showing sound wave propagation un-
der the NLSE dynamics in a disk with ε = 14and a perturbation in
the potential in section 4.1.5. . . . . . . . . . . . . . . . . . . . . . . 77
4.13 Trajectory of the vortex center when ε = 132
and time evolution of
dε1 for different ε for the motion of a single vortex in NLSE under
homogeneous Neumann BC with x01 = (0.35, 0.4) (left two) or x0
1 =
(0, 0.2) (right two) in (2.6) in section 4.2.1. . . . . . . . . . . . . . . . 78
4.14 Trajectory of the vortex pair (left), time evolution of Eε and Eεkin (sec-ond), xε1(t) and xε2(t) (third), and d
ε1(t) (right) in the NLSE dynamics
under homogeneous Neumann BC with ε = 132
and d0 = 0.5 in section
4.2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
List of Figures xvi
4.15 Trajectory and time evolution of xε1(t) and xε2(t) for d0 = 0.25 (top
left two), d0 = 0.7 (top right two) and d0 = 0.1 (bottom left two) and
time evolution of dε1(t) for d0 = 0.25 and d0 = 0.7 (bottom right two)
in section 4.2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.16 Trajectory of the vortex xε1 (blue line), xε2 (dark dash-dot line) and
xε3 (red dash line) and their corresponding time evolution for Case I
during small time (left column), intermediate time (middle column)
and large time (right column) with ε = 140
and d0 = 0.25 in section
4.2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.17 Contour plots of |ψε(x, t)| with ε = 116
at different times for the NLSE
dynamics of a vortex lattice for Case II with d1 = 0.6, d2 = 0.3 (top
two rows) and Case III with d1 = d2 = 0.3 (bottom two rows) in
section 4.2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.18 Contour plots of −|ψε(x, t)| (left) and slice plots of |ψε(0, y, t)| (right)at different times under the NLSE dynamics of a vortex cluster in Case
IV with d0 = 0.15 and ε = 140
for showing sound wave propagation in
section 4.2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.19 Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 4.2.5 . . . . . . . . . . . . . . . . 86
5.1 Trajectory of the vortex center in CGLE under Dirichlet BC when
ε = 132
for cases II-IV and VI and time evolution of dε1 for different ε
for cases II and VI (from left to right and then from top to bottom)
in section 5.1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Trajectory of the vortex center in CGLE under Dirichlet BC when
ε = 132
for cases IV-VII (left) and cases V-XII (right) in section 5.1.1. 90
5.3 Trajectory of the vortex center in CGLE under Dirichlet BC when
ε = 132
for cases: (a) I, XIII, XIV, (b) X, XV, XVI (from left to right)
in section 5.1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
List of Figures xvii
5.4 Trajectory of the vortex centers (a) and their corresponding time
evolution of the GL functionals (b) in CGLE dynamics under Dirichlet
BC when ε = 125
with different h(x) in (2.6) in section 5.1.2. . . . . . 92
5.5 Contour plot of |ψε(x, t)| for ε = 125
at different times as well as time
evolution of xε1(t) in CGLE dynamics and xr1(t) in the reduced dy-
namics under Dirichlet BC with h(x) = 0 in (2.6) and their difference
dε1(t) for different ε in section 5.1.2. . . . . . . . . . . . . . . . . . . . 94
5.6 Trajectory of the vortex centers (a) and their corresponding time
evolution of the GL functionals (b) in CGLE dynamics under Dirichlet
BC when ε = 125
with different h(x) in (2.6) in section 5.1.3. . . . . . 95
5.7 Contour plots of |ψε(x, t)| for ε = 125
at different times as well as time
evolution of xε1(t) in CGLE dynamics, xr1(t) in the reduced dynamics
under Dirichlet BC with h(x) = 0 in (2.6) and their difference dε1(t)
for different ε in section 5.1.3. . . . . . . . . . . . . . . . . . . . . . . 96
5.8 Trajectory of vortex centers for the interaction of different vortex
lattices in GLE under Dirichlet BC with ε = 132
and h(x) = 0 for
cases I-IX (from left to right and then from top to bottom) in section
5.1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.9 Contour plots of |φε(x)| for the steady states of vortex cluster in
CGLE under Dirichlet BC with ε = 132
for M = 8, 12, 16, 20 (from
left column to right column) and different domains: (a) unit disk
D = B1(0), (b) square domain D = [−1, 1]2, (c) rectangular domain
D = [−1.6, 1.6]× [−0.8, 0.8]. . . . . . . . . . . . . . . . . . . . . . . . 99
5.10 Contour plots of |φε(x)| for the steady states of vortex cluster in
CGLE under Dirichlet BC with ε = 132
and M = 12 on a unit disk
D = B1(0) (top row) or a square D = [−1, 1]2 (middle row) or a
rectangular domain D = [−1.6, 1.6]× [−0.8, 0.8] (bottom row) under
different h(x) = x+ y, x2 − y2, x− y, x2 − y2 + 2xy, x2 − y2 − 2xy
(from left column to right column). . . . . . . . . . . . . . . . . . . . 100
List of Figures xviii
5.11 Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 5.1.6 . . . . . . . . . . . . . . . . 101
5.12 Trajectory of the vortex center when ε = 125
(left) as well as time
evolution of xε1 (middle) and dε1 for different ε (right) for the motion
of a single vortex in CGLE under homogeneous Neumann BC with
different x01 in (2.6) in section 5.2.1.: (a) x0
1 = (0.1, 0), (b) x01 =
(0.1, 0.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.13 Contour plots of |ψε(x, t)| at different times when ε = 125
((a) &
(b)) and the corresponding time evolution of the GL functionals ((c)
& (d)) for the motion of vortex pair in CGLE under homogeneous
Neumann BC with different d0 in (2.6) in section 5.2.2: top row:
d0 = 0.3, bottom row: d0 = 0.7. . . . . . . . . . . . . . . . . . . . . . 103
5.14 Trajectory of the vortex center when ε = 125
(left) as well as time
evolution of xε1 (middle) and dε1 for different ε (right) for the motion of
vortex pair in CGLE under homogeneous Neumann BC with different
d0 in (2.6) in section 5.2.2: (a) d0 = 0.3, (b) d0 = 0.7. . . . . . . . . 104
5.15 Contour plots of |ψε(x, t)| at different times when ε = 125
and the
corresponding time evolution of the GL functionals for the motion
of vortex dipole in CGLE under homogeneous Neumann BC with
different d0 in (2.6) in section 5.2.3: top row: d0 = 0.3, bottom row:
d0 = 0.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.16 Trajectory of the vortex center when ε = 125
(left) as well as time
evolution of xε1 (middle) and dε1 for different ε (right) for the motion
of vortex dipole in CGLE under homogeneous Neumann BC with
different d0 in (2.6) in section 5.2.2: (a) d0 = 0.3, (b) d0 = 0.7. . . . 107
5.17 Trajectory of vortex centers for the interaction of different vortex
clusters in CGLE under Neumman BC with ε = 132
for cases I-IX
(from left to right and then from top to bottom) in section 5.2.4. . . 109
List of Figures xix
5.18 Contour plots of |ψε(x, t)| for the initial data ((a) & (c)) and corre-
sponding steady states ((b) & (d)) of vortex cluster in CGLE dynam-
ics under Neumman BC with ε = 132
and for cases I, III, V, VI, VII
and XIV (from left to right and then from top to bottom) in section
5.2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.19 Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 5.2.5 . . . . . . . . . . . . . . . . 112
5.20 Trajectory and time evolution of the distance between the vortex
center different ε for case I-III ((a)-(c)) in section 5.3. . . . . . . . . 113
6.1 Cartesian (or Eulerian) coordinates (x, y) (solid) and rotating La-
grangian coordinates (x, y) (dashed) in 2D for any fixed t ≥ 0. . . . . 127
6.2 The bounded computational domain D in rotating Lagrangian coor-
dinates x (left) and the corresponding domain A(t)D in Cartesian
(or Eulerian) coordinates x (right) when Ω = 0.5 at different times:
t = 0 (black solid), t = π4(cyan dashed), t = π
2(red dotted) and
t = 3π4(blue dash-dotted). The two green solid circles determine two
disks which are the union (inner circle) and the intersection of all do-
mains A(t)D for t ≥ 0, respectively. The magenta area is the vertical
maximal square inside the inner circle. . . . . . . . . . . . . . . . . . 132
6.3 Results for γx = γy = 1. Left: trajectory of the center of mass,
xc(t) = (xc(t), yc(t))T for 0 ≤ t ≤ 100. Right: coordinates of the
trajectory xc(t) (solid line: xc(t), dashed line: yc(t)) for different
rotation speed Ω, where the solid and dashed lines are obtained by
directly simulating the GPE and ‘*’ and ‘o’ represent the solutions to
the ODEs in Lemma 6.2.3. . . . . . . . . . . . . . . . . . . . . . . . . 144
List of Figures xx
6.4 Results for γx = 1, γy = 1.1. Left: trajectory of the center of mass,
xc(t) = (xc(t), yc(t))T for 0 ≤ t ≤ 100. Right: coordinates of the
trajectory xc(t) (solid line: xc(t), dashed line: yc(t)) for different
rotation speed Ω, where the solid and dashed lines are obtained by
directly simulating the GPE and ‘*’ and ‘o’ represent the solutions to
the ODEs in Lemma 6.2.3. . . . . . . . . . . . . . . . . . . . . . . . . 145
6.5 Time evolution of the angular momentum expectation (left) and en-
ergy and mass (right) for Cases (i)-(iv) in section 5.3. . . . . . . . . . 146
6.6 Time evolution of condensate widths in the Cases (i)–(iv) in section
5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.7 Contour plots of the density function |ψ(x, t)|2 for dynamics of a vor-
tex lattice in a rotating BEC (Case (i)). Domain displayed: (x, y) ∈[−13, 13]2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.8 Contour plots of the density function |ψ(x, t)|2 for dynamics of a vor-
tex lattice in a rotating dipolar BEC (Case (ii)). Domain displayed:
(x, y) ∈ [−10, 10]2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
List of Symbols and Abbreviations
2D two dimension
3D three dimension
BEC Bose-Einstein condensate
GLSE Ginzburg–Landau–Schrodinger equation
GLE Ginburg–Landau equation
NLSE Nonlinear Schrodinger equation
CGLE complex Ginburg–Landau equation
GPE Gross–Pitaevskii equation
RDL reduced dynamical law
BC boundary condition
CNFD Crank–Nicolson finite difference
TSCNFD time–splitting Crank–Nicolson finite difference
TSCP time–splitting cosine pseudospectral
FEM finite element method
SAM surface adiabatic model
SDM surface density model
Fig. figure
~ Planck constant
xxi
List of Symbols and Abbreviations xxii
(r, θ) polar coordinate
∇ gradient
∆ = ∇ · ∇ Laplacian
x Cartesian coordinate
x rotating Lagrangian coordinate
τ time step size
h space mesh size
i imaginary unit
f Fourier transform of function f
f ∗ conjugate of of a complex function f
f ∗ g convolution of function f with function g
Re(f) real part of a complex function f
Im(f) imaginary part of a complex function f
Ω angular velocity
ωx, ωy, ωz trapping frequencies in x-, y-, and z- direction
Lz = −i(x∂y − y∂x) z-component of angular momentum
〈Lz〉(t) angular momentum expectation
xc(t) center of mass of a condensate
σα(t) (α = x, y, or z) condensate width in α-direction
ψ(x, t), ψε(x, t) macroscopic wave function
φ(x, t), φεm(x, t) stationary state
ρε(x, t) = |φε(x, t)|2 position density
Sε(x, t) = Arg(φε(x, t)) phase of the wave function
Chapter 1
Introduction
Vortex, which can exist in vast areas, is any spiral motion with closed stream
lines. It can survive not only in macro scale such as in the air, liquid or the tur-
bulent flow, but also in micro scale such as the Bose-Einstein condensate (BEC),
the superfluidity and superconductivity, etc. The micro-vortices differ from those
macro-vortices by the so-called ‘vorticity’, which is a mathematical concept related
to the amount of ‘circulation’ or ‘rotation’. Among those micro-vortices, the quan-
tized vortex that arises from quantum mechanics distinguish itself from others by
the signature of ‘quantized vorticity’.
1.1 Vortex in superfluidity and superconductivity
Quantized vortices are topological defects that arise from the order parameter
in superfluids, Bose-Einstein condensate (BEC) and superconductors in which fric-
tionless fluids flow with circulation being quantized around each vortex.
Bose-Einstein condensation, superconductivity and superfluidity are among the
most intriguing phenomena in nature. Their astonishing properties are direct con-
sequences of quantum mechanics. While most other quantum effects only appear
in matter on the atomic or subatomic scale, superfluids and superconductors show
the effects of quantum mechanics acting on the bulk properties of matter on a large
1
1.1 Vortex in superfluidity and superconductivity 2
scale. They are macroscopic quantum phenomena. This is an essential origin of su-
perfluidity and superconductivity, in which macroscopically phase coherence allows
a dissipationless current to flow. Bulk superfluids are distinguished from normal
fluids by their ability to support dissipationless flow.
Superconductivity is a phenomenon of exactly zero electrical resistance occurring
in certain materials at low temperature. It was discovered by Heike Kamerlingh
Onnes in 1911. Type-I superconductivity is characterized by the so-called Meissner
effect, which introduce the complete exclusion of magnetic from the superconductor.
While for the type-II superconductors in the so-called mixed vortex state, quantized
amount of magnetic flux carried by the vortex lines is allowed to penetrate the
superconductors [58, 60].
A Bose-Einstein condensate (BEC) is a state of matter of a dilute gas of weakly
interacting bosons below some critical temperature. It supports the quantum effects
in macroscopic scale since numbers of the bosons will condense into the single-
particle state, at which point we can treat those condensed bosons as one-particle
[2,88,127,130,135]. The phenomena of BEC was predicted in 1924 by Albert Einstein
based on the work of Satyendra Bath Bose and was first realized in experiments in
1955 [7, 39, 52]. Later, with the observation of quantized vortices [2, 40, 109, 110,
112, 128, 156], plenty of work have been devoted to study the phenomenological
properties of vortices in the rotating BEC, dipolar BEC, multi-component BEC and
spinor BEC, etc, which has now opened the door to the study of superfluidity in the
Bose-system [4, 91].
Superfluid is a state of matter characterized by the complete absence of viscosity.
In other words, if placed in a closed loop, superfluids can flow endlessly without
friction. Known as a major facet in the study of quantum hydrodynamics, the
superfluidity effect was discovered by Kapitsa, Allen and Misener in 1937. The
formation of the superfluid is known to be related to the formation of a BEC. This
is made obvious by the fact that superfluidity occurs in liquid helium-4 at far higher
temperatures than it does in helium-3. Each molecule of helium-4 is a boson particle,
1.2 Problems and contemporary studies 3
by virtue of its zero spin. Helium-3, however, is a fermion particle, which can form
bosons only by pairing with itself at much lower temperatures, in a process similar
to the electron pairing in superconductivity.
Feynman [63] predicted that the rotation of superfluids might be subject to the
quantized vortices in 1955, while in 1957 Abrikosov [3] predicted the existence of
the vortex lattice in superconductors. Studies on phenomena related to quantized
vortex has since boomed and the Nobel Prize in Physics was recently awarded to
Cornell, Weimann and Ketterle in 2001 for their decisive contributions to Bose-
Einstein condensation and to Ginzburg, Abrikosov and Leggett in 2003 for their
pioneering contributions to superfluidity and superconductivity.
1.2 Problems and contemporary studies
In recent years, phenomenological properties of quantized vortices in superflu-
idity and superconductivity have been extensively studied by both mathematical
analysis and numerical simulations. It is remarkable that many of those properties
can be well characterized by relatively simple models such as the Ginzburg-Landau-
Schrodinger equation (GLSE) [12] and the Gross-Pitaesvkii equation (GPE) [19,127].
In this thesis, we focus on the following two subjects.
1.2.1 Ginzburg-Landau-Schrodinger equation
First, we are concerned with the vortex dynamics and interactions in a specific
form of 2D Ginzburg-Landau-Schrodinger equation , which describe a vast variety of
phenomena in physics community, ranging from superconductivity and superfluidity
to strings in field theory, from the second order phase transition to nonlinear waves
[12, 64, 66, 87, 126, 129]:
(λε + iβ)∂tψε(x, t) = ∆ψε +
1
ε2(V (x)− |ψε|2)ψε, x ∈ D, t > 0, (1.1)
1.2 Problems and contemporary studies 4
with initial condition
ψε(x, 0) = ψε0(x), x ∈ D, (1.2)
and under either Dirichlet boundary condition (BC)
ψε(x, t) = g(x) = eiω(x), x ∈ ∂D, t ≥ 0, (1.3)
or homogeneous Neumann BC
∂ψε(x, t)
∂ν= 0, x ∈ ∂D, t ≥ 0. (1.4)
Here, D ⊂ R2 is a smooth and bounded domain, t is time, x = (x, y) ∈ R2 is
the Cartesian coordinate vector, V (x) satisfying lim|x|→∂D V (x) = 1 is a positive
real-valued smooth function, ψε := ψε(x, t) is a complex-valued wave function (or-
der parameter), ω is a given real-valued function, ψε0 and g are given smooth and
complex-valued functions satisfying the compatibility condition ψε0(x) = g(x) for
x ∈ ∂D, ν = (ν1, ν2) and ν⊥ = (−ν2, ν1) ∈ R2 satisfying |ν| =√ν21 + ν22 = 1 are
the outward normal and tangent vectors along ∂D, respectively, i =√−1 is the
unit imaginary number, 0 < ε < 1 is a given dimensionless constant, and λε, β are
two nonnegative constants satisfying λε + β > 0. The GLSE covers many different
equations arise in various different physical fields. For example, when λε 6= 0, β = 0,
it reduces to the Ginzburg-Landau equation (GLE) for modelling superconductiv-
ity. When λε = 0, β = 1, the GLSE collapses to the nonlinear Schrodinger equation
(NLSE) which is well known for modelling, for example, BEC or superfluidity. While
λε > 0 and β > 0, the GLSE is the so-called complex Ginzburg-Landau equation
(CGLE) or nonlinear Schrodinger equation with damping term which arise in the
study of the hall effect in type II superconductor.
In superconductivity, V (x) ≡ 1 stands for the equilibrium density of supercon-
ducting electron [44, 45, 57]. When V (x) ≡ 1, the medium is uniform, while if
V (x) 6≡ 1, the medium is inhomogeneous which is used to, for example, describe the
pining effect in superconductor with impurities.
1.2 Problems and contemporary studies 5
Denote the Ginzburg-Landau (GL) functional (‘energy’) as [48, 76, 107]
Eε(t) :=∫
D
[1
2|∇ψε|2 + 1
4ε2(V (x)− |ψε|2
)2]dx = Eεkin(t) + Eεint(t), t ≥ 0, (1.5)
whose corresponding Euler–Lagrange equation reads as:
∆ψε +1
ε2(V (x)− |ψε|2)ψε = 0, x ∈ D. (1.6)
In (1.5), the kinetic and interaction energies are defined as
Eεkin(t) :=1
2
∫
D|∇ψε|2dx, Eεint(t) :=
1
4ε2
∫
D
(V (x)− |ψε|2
)2dx, t ≥ 0,
respectively. The GLSE (1.1) now can be rewritten as
(λε + iβ)∂tψε(x, t) = −δE(ψ)
δψ∗ , (1.7)
where ψ∗ denotes the complex conjugate of function ψ. Moreover, it is easy to show
that the GLE or CGLE dissipates the total energy, i.e., dEε
dt≤ 0, while the NLSE
conserve the total energy, i.e., dEε
dt= 0.
During the last several decades, constructions and analysis of the solutions of
(1.6) as well as vortex dynamics and interaction related to the GLSE (1.1) under
different scalings have been extensively studied in the literatures.
For GLE defined in R2, under the normal scaling λε = ε ≡ 1 and homogeneous
potential V (x) ≡ 1, Neu [118] found numerically that quantized vortices with wind-
ing number m = ±1 are dynamically stable, and respectively, |m| > 1 dynamically
unstable. Based on the assumption that the vortices are well separated and of wind-
ing number +1 or −1, he also obtained formally the reduced dynamical law (RDL)
governing the motion of the vortex centers by method of asymptotic analysis. How-
ever, this RDL is only correct up to the first collision time and cannot indicate the
motion of multi-degree vortices. Recently, in a series of papers [32, 34, 35], Bethuel
et al. investigated the asymptotic behaviour of vortices as ε → 0 under the accel-
erating time scale λε =1
ln 1ε
. Under very general assumptions (which release those
constrains in Neu’s work), they proved that the limiting vortices, which can be of
1.2 Problems and contemporary studies 6
multiple degree, move according to a RDL, which is a set of simple ordinary differ-
ential equations (ODEs). Much stronger than Neu’s RDL, this RDL is always valid
except for a finite number of times that representing vortex splittings, recombina-
tions and/or collisions. Their studies also show an interesting phenomena called as
“phase-vortex interaction”, the phenomena that can cause an unexpected drift of
the vortices, which they pointed out that cannot occur in the case of the domain be-
ing bounded. Moreover, they conducted some similar research in higher dimensional
space [33].
In the bounded domain case when the potential is homogeneous, i.e., V (x) ≡ 1
Lin [99, 100, 102] extended Neu’s results by considering the dynamics of vortices
in the asymptotic limit ε → 0 under various scales of λε and with different BCs.
Based on the well-preparation assumption similar to Neu’s, he derived the RDLs
that govern the motion of these vortices and rigorously proved that vortices move
with velocities of the order of | ln ε|−1 if λε = 1. Similar studies have also been
conducted by E [61], Jerrard et al. [75], Jimbo et al. [82,85] and Sandier et al. [134].
Unfortunately, all those RDLs are only valid up to the first time that the vortices
collide and/or exit the domain and cannot describe the motion of multiple degree
vortices. Recently, Serfaty [138] extended the RDL of the vortices after collisions,
but still under the assumption that those vortices are of degree +1 or −1 and that
only simple collision could happen during dynamics (i.e, the situation that more
than two vortices meet at the same time and place are not allowed). Actually, the
motion of the multiple degree vortices and the dynamics of vortices after collision
and/or splittings still remain as interesting open problems. When the potential is
inhomogeneous, i.e., V (x) 6≡ 1, Jian et al. [77–79] investigated the pinning effect
of the vortices asymptotically as ε → 0 in the GLE with Dirichlet BC under the
scale λε = 1. They established the corresponding RDLs that govern the dynamics
of limiting vortices.
As for the steady states of GLE or the solution of Euler–Lagrange equation (1.6),
situations are quite different case by case. In the whole plane case, as indicated by
1.2 Problems and contemporary studies 7
Neu’s results [118], it was generally believed that two vortices with winding number
of opposite sign undergo attractive interaction and tend to coalesce and annihilation.
Hence, for the steady states of the GLE in whole plane, either there are no vortices
or all the vortices are of the same sign. However, when the domain is bounded,
Lin [101] proved the existence of the mixed vortex-antivortex solution of the Euler–
Lagrange equation subject to the Dirichlet BC (1.3) for sufficiently small ε, i.e., the
steady states of GLE under Dirichlet BC (1.3) allows vortices with winding number
of opposite sign. Nevertheless, Jimbo et al. [83] and Serfaty [137] obtained that any
solutions with vortices to (1.1) and (1.4) are unstable in a convex or simple connected
domain, while recently del Pino et al. [53] proved the existence of the solution with
exactly k vortices of degree one for any integer number k if the domain were not
simply connected by the approach of variational reduction. Hence, all the vortices
in the initial data (1.2) will either collide with each other and annihilate or simply
exit the domain finally. Actually, several studies had been established in both the
planar domains and/or higher dimensional domains for the stability of the steady
state solution of GLE with Neumann BC (1.4) [51, 81, 83, 84, 86], which imply the
close relation between the stability of the equilibrium solution with vortices and the
geometrical property of the domain.
For NLSE defined in R2, when V (x) = 1 and ε = 1, Bethuel et al. [36] proved
global well-posedness of NLSE for classes of initial data that have vortices. For
the vortex dynamics, Fetter [62] predicted that, to the leading order, the motion
of vortices in the NLSE would be governed by the same law as that in the ideal
incompressible fluid. Then, the same prediction was given by Neu [118]. He conjec-
tured the stability of the vortex states under NLSE dynamics as an open problem,
based on which he found that the vortices behave like point vortices in ideal fluid,
and obtained the corresponding RDLs. However, these RDLs are only correct up to
the leading order. Corrections to this leading order approximation due to radiation
and/or related questions when long-time dynamics of vortices is considered still re-
main as important open problems. In fact, using the method of effective action and
1.2 Problems and contemporary studies 8
geometric solvability, Ovichinnikov and Sigal confirmed Neu’s approximation and
derived some leading radiative corrections [121, 122] based on the assumption that
the vortices are well separated, which was extended by Lange and Schroers [98] to
study the dynamics of overlapping vortices. Recently, Bethuel et al. [31] derived the
asymptotic behaviour of the vortices as ε→ 0.
In the bounded domain case, when V (x) = 1, many papers have been dedicated
to the study of the vortex states and dynamics after Neu’s work [118]. Mironescu
[115] investigated stability of the vortices in NLSE with (1.3) and showed that for
fixed winding number m: a vortex with |m| = 1 is always dynamical stable; while
for those of winding number |m| > 1, there exists a critical εcm such that if ε > εcm,
the vortex is stable, otherwise unstable. Mironescu’s results were then improved
by Lin [103] using the spectrum of a linearized operator. Subsequently, Lin and
Xin [107] studied the vortex dynamics on a bounded domain with either Dirichlet
or Neumann BC, which was further investigated by Jerrard and Spirn [76]. In
addition, Colliander and Jerrard [48,49] studied the vortex structures and dynamics
on a torus or under periodic BC. In these studies, the authors derived the RDLs
which govern the dynamics of vortex centers under the NLSE dynamics when ε→ 0
with fixed distances between different vortex centers initially. They obtained that to
the leading order the vortices move according to the Kirchhoff law in the bounded
domain case. However, these reduced dynamical laws cannot indicate radiation
and/or sound propagations created by highly co-rotating or overlapping vortices. In
fact, it remains as a very fascinating and fundamental open problem to understand
the vortex-sound interaction [119], and how the sound waves modify the motion of
vortices [64].
For the CGLE under scaling λε =1
ln 1ε
and homogeneous potential, based on some
proper assumptions, Miot [114] studied the dynamics of vortices asymptotically as
ε → 0 in the whole plane case while Kurzke et al. [95] investigated that in the
bounded domain case, the corresponding RDLs were derived to govern the motion
of the limiting vortices in the whole plane and/or the bounded domain, respectively.
1.2 Problems and contemporary studies 9
The results shows that the RDLs in the CGLE is actually a hybrid of RDL for
GLE and that for NLSE. More recently, Serfaty and Tice [139] studied the vortex
dynamics in a more complicated CGLE which involves electromagnetic field and
pinning effect.
On the numerical aspects, finite element methods were proposed to investigate
numerical solutions of the Ginzburg-Landau equation and related Ginzburg-Landau
models of superconductivity [5, 46, 56, 60, 89]. Recently, by proposing efficient and
accurate numerical methods for discretizing the GLSE in the whole space, Zhang
et al. [160, 161] compared the dynamics of quantized vortices from the reduced
dynamical laws obtained by Neu with those obtained from the direct numerical
simulation results from GLE and/or NLSE under different parameters and/or initial
setups. They solved numerically Neu’s open problem on the stability of vortex
states under the NLSE dynamics, i.e., vortices with winding number m = ±1 are
dynamically stable, and resp., |m| > 1 dynamically unstable [160,161], which agree
with those derived by Ovchinnikov and Sigal [120]. In addition, they identified
numerically the parameter regimes for quantized vortex dynamics when the reduced
dynamical laws agree qualitatively and/or quantitatively and fail to agree with those
from GLE and/or NLSE dynamics.
However, to our limited knowledge, there were few numerical studies on the
vortex dynamics and interaction of the GLSE (1.1) in bounded domain, much less
for the sound-vortex interaction in the NLSE dynamics.
1.2.2 Gross-Pitaevskii equation with angular momentum
The occurrence of quantized vortices is a hallmark of the superfluid nature
of Bose–Einstein condensates. In addition, condensation of bosonic atoms and
molecules with significant dipole moments whose interaction is both nonlocal and
anisotropic has recently been achieved experimentally in trapped 52Cr and 164Dy
gases [1, 50, 69, 97, 108, 111, 151].
Using the mean field approximation, when the temperature T is much smaller
1.2 Problems and contemporary studies 10
than the critical temperature Tc, the properties of a BEC in a rotating frame with
long-range dipole-dipole interaction are well described by the macroscopic complex-
valued wave function ψ = ψ(x, t), whose evolution is governed by the following
three-dimensional (3D) Gross-Pitaevskii equation (GPE) with angular momentum
rotation term and long-range dipole-dipole interaction [1, 17, 41, 136, 148, 152, 162]:
i~∂tψ(x, t) =
[− ~2
2m∇2 + V (x) + U0|ψ|2 +
(Vdip ∗ |ψ|2
)− ΩLz
]ψ(x, t), t > 0, (1.8)
where t denotes time, x = (x, y, z)T ∈ R3 is the Cartesian coordinate vector, ~ is
the Planck constant, m is the mass of a dipolar particle and V (x) is an external
trapping potential, which reads as
V (x) =m
2(ω2
xx2 + ω2
yy2 + ω2
zz2) (1.9)
if a harmonic trap potential is concerned with. Here, ωx, ωy and ωz are the trap fre-
quencies in x-, y- and z-directions, respectively. U0 =4π~2asm
represents short-range
(or local) interaction between dipoles in the condensate with as the s-wave scatter-
ing length. Vdip(x) describes the long-range dipolar interaction potential between
dipoles, which is defined as
Vdip(x) =µ0µ
2dip
4π
1− 3(x · n)2/|x|2|x|3 =
µ0µ2dip
4π
1− 3 cos2(ϑ)
|x|3 , x ∈ R2,
where µ0 and µdip are the vacuum permeability and permanent magnetic dipole
moment, respectively (e.g., µdip = 6µBfor 52Cr with µB
being the Bohr magneton),
n = (n1, n2, n3)T ∈ R3 is a given unit vector, i.e., |n| =
√n21 + n2
2 + n23 = 1,
representing the dipole axis (or dipole moment) and ϑ = ϑn(x) is the angle between
the dipole axis n and the vector x. In addition, Ω is the angular velocity of the laser
beam and Lz = −i~(x∂y − y∂x) is the z-component of the angular momentum L =
x × P with the momentum operator P = −i~∇. The wave function is normalized
to
||ψ||22 :=∫
R3
|ψ(x, t)|2dx = N,
1.2 Problems and contemporary studies 11
with N being the total number of dipolar particles in the dipolar BEC. Introducing
the dimensionless variables, t → t/ω0 with ω0 = minωx, ωy, ωz, x → a0x and
ψ →√Nψ/a
320 , we have the dimensionless rotational dipolar GPE [19, 152, 153]:
i∂tψ(x, t) =
[−1
2∇2 + V (x) + κ|ψ|2 + λ
(Udip ∗ |ψ|2
)− ΩLz
]ψ(x, t), (1.10)
where κ = 4πNasxs
, λ =mNµ0µ2dip
3~2xs, V (x) = 1
2(γ2xx
2 + γ2yy2 + γ2zz
2) is the dimensionless
harmonic trapping potential with γx = ωx/ω0, γy = ωy/ω0, γz = ωz/ω0, and Udip is
the dimensionless long-range dipole-dipole interaction potential defined as
Udip(x) =3
4π|x|3[1− 3(x · n)2
|x|2]=
3
4π|x|3[1− 3 cos2(ϑ)
], x ∈ R3. (1.11)
The wave function is normalized to
‖ψ‖2 :=∫
R3
|ψ(x, t)|2 dx = 1. (1.12)
In addition, similar to [17, 41], the above GPE (1.10) can be re-formulated as the
following Gross-Pitaevskii-Poisson system [14, 17, 41]
i∂tψ(x, t) =
[−1
2∇2 + V (x) + (κ− λ)|ψ|2 − 3λϕ(x, t)− ΩLz
]ψ(x, t), (1.13)
ϕ(x, t) = ∂nnu(x, t), −∇2u(x, t) = |ψ(x, t)|2 with lim|x|→∞
u(x, t) = 0, (1.14)
where ∂n = n · ∇ and ∂nn = ∂n(∂n). From (1.14), it is easy to see that for t ≥ 0
u(x, t) =
(1
4π|x|
)∗ |ψ|2 :=
∫
R3
1
4π|x− x′| |ψ(x′, t)|2dx′, x ∈ R3. (1.15)
Recently, many numerical and theoretical studies have been done on rotating
(dipolar) BECs. There have been many numerical methods proposed to study the
dynamics of non-rotating BECs, i.e. when Ω = 0 and λ = 0 [5,19,25,42,90,116,144].
Among them, the time-splitting sine/Fourier pseudospectral method is one of the
most successful methods. It has spectral accuracy in space and is easy to implement.
In addition, as shown in [17], this method can also be easily generalized to simulate
the dynamics of dipolar BECs when λ 6= 0. However, in rotating condensates, i.e.,
when Ω 6= 0, we can not directly apply the time-splitting pseudospectral method
1.3 Purpose and scope of this thesis 12
proposed in [25] to study their dynamics due to the appearance of angular rotational
term. So far, there have been several methods introduced to solve the GPE with an
angular momentum term. For example, a pseudospectral type method was proposed
in [18] by reformulating the problem in the two-dimensional polar coordinates (r, θ)
or three-dimensional cylindrical coordinates (r, θ, z). The method is of second-order
or fourth-order in the radial direction and spectral accuracy in other directions. A
time-splitting alternating direction implicit method was proposed in [24], where the
authors decouple the angular terms into two parts and apply the Fourier transform in
each direction. Furthermore, a generalized Laguerre-Fourier-Hermite pseudospectral
method was presented in [21]. These methods have higher spatial accuracy compared
to those in [5, 16, 90] and are also valid in dissipative variants of the GPE (1.10),
cf. [147]. On the other hand, the implementation of these methods can become quite
involved.
1.3 Purpose and scope of this thesis
As shown in the last two subsections, a vast number of researches have been
done and plenty of results have been obtained for the vortex dynamics in BEC,
superfluidity and superconductivity. However, there are still some limitations.
• For the vortex dynamics in superconductivity and superfluidity on bounded
domain, most studies are primarily researches of the RDLs of well separated
vortices. Vortex phenomena related to overlapping vortices and/or vortex col-
lision as well as the effect of the boundary condition and effect of the domain
geometry on the vortex dynamics still remains unknown. Numerical simula-
tions have become powerful and useful to figure out those exotic phenomena.
However, few numerical studies for the bounded domain case were reported.
• For the vortex dynamics in GPE with angular momentum, there have been
only a few reports about the interactions between a few vortices. Moreover,
the existing numerical methods have their own limitations. (i). The finite
1.3 Purpose and scope of this thesis 13
difference method (FDM) or finite element method (FEM) usually need a
very fine mesh size, and their order of accuracy are usually low, hence they are
time–consuming and inefficient. (ii). Although time splitting spectral method
with alternative direction technique is of spectral accuracy, they might cause
some problems when the rotating frequency is large. (iii). Additionally, the
generalized Laguerre-Fourier-Hermite pseudospectral method is not easy to
implement.
Hence, in this thesis, we mainly focus on the following two parts:
• (i). to present efficient and accurate numerical methods for discretizing the
reduced dynamical laws and the GLSE (1.1) on bounded domains under dif-
ferent BCs, (ii). to understand numerically how the boundary condition and
radiation as well as geometry of the domain affect vortex dynamics and in-
teraction, (iii). to investigate the pining effect of the vortices in CGLE and
GLE dynamics, (iv). to study numerically vortex interaction in the GLSE
dynamics and/or compare them with those from the reduced dynamical laws
with different initial setups and parameter regimes, and (v). to identify cases
where the reduced dynamical laws agree qualitatively and/or quantitatively
as well as fail to agree with those from GLSE on vortex interaction.
• to propose a simple and efficient numerical method to solve the GPE with
angular momentum rotation term which may include a dipolar interaction
term. One novel idea in this method consists in the use of rotating Lagrangian
coordinates as in [11] in which the angular momentum rotation term vanishes.
Hence, we can easily apply the method for non-rotating BECs in [25] to solve
the rotating case.
Studies for the first part will be carried out in chapter 2 to chapter 5, while research
on the second part will be conducted in chapter 6. In chapter 7, conclusions and
possible directions of future work will be summarized and discussed.
Chapter 2
Methods for GLSE on bounded domain
In this chapter, begin with the stationary vortex state of the Ginzburg-Landau-
Schrodinger (GLSE) equation, various RDLs that governed the motion of the vortex
centers under different boundary conditions (BCs) are reviewed and their equivalent
forms are presented and proved. Then, accurate and efficient numerical methods are
proposed for computing the GLSE in a disk or rectangular domain under Dirichlet
or homogeneous Neumann BC. These methods will be applied to study various
phenomena on the vortex dynamics and interaction in following chapters.
2.1 Stationary vortex states
To consider the vortex solution of the GLSE (1.1), we consider the following time
independent GLSE with V (x) = 1 in a disk domain centered at origin with radius
R0, i.e., D = BR0(0):
∆φε +1
ε2(1− |φε|2)φε = 0, x ∈ D, (2.1)
|φε(x, t)| = 1, if x ∈ ∂D, (2.2)
where φε(x, t) is a complex-valued function which can be viewed as the steady states
of the GLSE (1.1) in a disk domain. The vortex solution takes the form of:
φεm(x) = f εm(r)eimθ, x = (r cos(θ), r sin(θ)) ∈ D, (2.3)
14
2.1 Stationary vortex states 15
0 0.25 0.50
0.5
1
r
f m ε
0 0.25 0.50
0.5
1
r
f m ε
m=1m=2m=3m=4m=5
ε=1/25ε=1/32ε=1/40ε=1/50ε=1/64
Figure 2.1: Plot of the function f εm(r) in (2.4) with R0 = 0.5. left: ε = 140
with
different winding number m. right: m = 1 with different different ε.
whose existence and qualitative properties were carried out in [71,72]. Here, m ∈ Z
is called as the topological charge or winding number or index that represents the
singularity of the vortex, the modulus f εm(r) is a real-valued function satisfying:
[1
r
d
dr
(rd
dr
)− 1
r2+
1
ε2(1− (f εm(r))
2)]f εm(r) = 0, 0 < r < R0, (2.4)
f εm(r = 0) = 0, f εm(r = R0) = 1. (2.5)
Numerically, the solution f εm can be obtained by either employing a shooting method
[47] or a finite difference method with Newton iteration being used for the resulted
non-linear system [160]. Fig. 2.1 depicts the results for function f εm(r) with different
ε and m, while Fig. 2.2 shows the surf plots of the density |φεm|2 and the contour
plots of the corresponding phase for m = 1 and m = 5. The stability of the vortex
was investigated by Mironescu [115]. He showed that for fixed winding number m,
the vortex with |m| = 1 is always dynamical stable while for those of winding number
|m| > 1, there is a critical εcm such that if ε > εcm, the vortex is stable, otherwise
unstable. Mironescu’s results was then improved by Lin [103] by considering the
spectrum of a linearized operator. It might be interesting to study how the stability
of a vortex depends on the perturbation, and how the vortices of high index split if
they are not stable.
2.2 Reduced dynamical laws 16
(a)
(b)
Figure 2.2: Surf plot of the density |φεm|2 (left column) and the contour plot of the
corresponding phase (right column) for m = 1 (a) and m = 4 (b).
2.2 Reduced dynamical laws
It had been pointed out that the vortex of index |m| = 1 is always stable, and
those of index |m| > 1 is stable only up to some condition. Thus, it should be
interesting to understand how those vortices of winding number |m| = 1 dynamic
and interact with each other, and how the BC, the geometry of the domain affect
their motion. To this end, we choose the initial data in (1.2) as:
ψε0(x) = eih(x)M∏
j=1
φεnj(x− x0
j ) = eih(x)M∏
j=1
φεnj(x− x0j , y − y0j ), x ∈ D, (2.6)
2.2 Reduced dynamical laws 17
here M > 0 is the total number of vortices in the initial data, the phase shift h(x) is
a harmonic function and for j = 1, 2, . . . ,M , nj = 1 or −1, and x0j = (x0j , y
0j ) ∈ D are
the winding number and initial location of the j-th vortex, respectively. Moreover,
φεnjis chosen as
φεnj=
f εnj(|x|)einjθ(x), if 0 ≤ |x| ≤ R0,
einjθ(x), if |x| ≥ R0,(2.7)
where f εnjis the modulus of the vortex solution in (2.3) with winding number nj
and R0 is constant which is small than the diameter of the domain D.
It is well known that to the leading order in the limit ε → 0, theM well separated
vortices move according to the reduced dynamical law, which are ODE systems. In
this section, we review various reduced dynamical laws in different cases and present
some equivalent forms. We divide into three parts. The first part and second part
are devoted to the case of the GLSE (1.1) under Dirichlet and/or homogeneous BC
without pinning effect, i.e., V (x) ≡ 1, respectively. The third part is concerned with
the GLE with inhomogeneous potential, i.e., V (x) 6≡ 1 under Dirichlet BC.
2.2.1 Under homogeneous potential
In this section, we let λε =α
ln(1/ε). To simplify our presentation, for j = 1, · · · , N ,
hereafter we let xεj(t) = (xεj(t), yεj (t)) be the location of the M distinct and isolated
vortex centers in the solution of the GLSE (1.1) with initial condition (2.6) at time
t ≥ 0, and denote
X0 := (x01,x
02, . . . ,x
0M), Xε := Xε(t) = (xε1(t),x
ε2(t), . . . ,x
εM(t)), t ≥ 0,
then we have [48, 74, 95, 100]:
Theorem 2.2.1. As ε→ 0, for j = 1, · · · , N , the vortex center xεj(t) will converge
to point xj(t) satisfying:
(αI + βnjJ)dxj(t)
dt= −∇xj
W (X), 0 ≤ t < T, (2.8)
xj(t = 0) = x0j . (2.9)
2.2 Reduced dynamical laws 18
In equation (2.8), T is the first time that either two vortex collide or any vortex exit
the domain, X := X(t) = (x1(t),x2(t), . . . ,xM(t)),
I =
1 0
0 1
, J =
0 −1
1 0
,
are the 2 × 2 identity and symplectic matrix, respectively. Moreover, the function
W (X) is the so called renormalized energy defined as:
W (X) =: Wcen(X) +Wbc(X), (2.10)
where Wcen is the renormalized energy associated to the M vortex centers that
defined as
Wcen(X) = −∑
1≤i 6=j≤Nninj ln |xi − xj|, (2.11)
and Wbc(X) is the renormalized energy involving the effect of the BC (1.3) and/or
(1.4), which takes different formations in different cases.
Under Dirichlet boundary condition
For the GLSE (1.1) with initial condition (2.6) under Dirichlet BC (1.3), it has
been derived formally and rigorously [30,48,95,102,106,138] thatWbc(X) = Wdbc(X)
in the renormalized energy (2.10) admits the form:
Wdbc(X) =: −M∑
j=1
njR(xj ;X) +
∫
∂D
R(x;X) +
M∑
j=1
nj ln |x− xj|
∂ν⊥ω(x)
2πds, (2.12)
where, for any fixed X ∈ DM , R(x;X) is a harmonic function in x, i.e.,
∆R(x;X) = 0, x ∈ D, (2.13)
satisfying the following Neumann BC
∂R(x;X)
∂ν= ∂ν⊥ω(x)−
∂
∂ν
M∑
l=1
nl ln |x− xl|, x ∈ ∂D. (2.14)
2.2 Reduced dynamical laws 19
Notice that to calculate ∇xjW (X), we need to calculate ∇xj
R, and since for j =
1, · · · , N , xj is implicitly included in R(x, X) as a parameter, hence it is difficult
to calculate ∇xjR and thus difficult to solve the reduced dynamical law (2.8) with
(2.10)–(2.12) even numerically. However, by using an identity in [30] (see Eq. (51)
on page 84),
∇xj[W (X) +Wdbc(X)] = −2nj∇x
[R(x;X) +
M∑
l=1&l 6=jnl ln |x− xl|
]
x=xj
,
we have the following simplified equivalent form for (2.8).
Lemma 2.2.1. For 1 ≤ j ≤M and t > 0, system (2.8) can be simplified as
(αI+βmjJ)d
dtxj(t) = 2nj
[∇xR (x;X) |x=xj(t) +
M∑
l=1&l 6=jnl
xj(t)− xl(t)
|xj(t)− xl(t)|2
]. (2.15)
Moreover, for any fixed X ∈ DM , by introducing function H(x, X) and Q(x, X)
that both are harmonic in x satisfying respectively the boundary condition [75,107]:
∂H(x;X)
∂ν⊥= ∂ν⊥ω(x)−
∂
∂ν
M∑
l=1
nl ln |x− xl|, x ∈ ∂D, (2.16)
Q(x;X) = ω(x)−M∑
l=1
nlθ(x− xl), x ∈ ∂D, (2.17)
with the function θ : R2 → [0, 2π) defined as
cos(θ(x)) =x
|x| , sin(θ(x)) =y
|x| , 0 6= x = (x, y) ∈ R2, (2.18)
we have the following lemma for the equivalence of the reduced dynamical law
(2.15) [22, 23]:
Lemma 2.2.2. . For any fixed X ∈ DM , we have the following identity
J∇xQ (x;X) = ∇xR (x;X) = J∇xH (x;X) , x ∈ D, (2.19)
which immediately implies the equivalence between system (2.15) and the following
2.2 Reduced dynamical laws 20
two systems: for t > 0
(αI + βnjJ)d
dtxj(t) = 2nj
[J∇xH (x;X) |x=xj(t) +
M∑
l=1&l 6=jnl
xj(t)− xl(t)
|xj(t)− xl(t)|2
],
(αI + βnjJ)d
dtxj(t) = 2nj
[J∇xQ (x;X) |
x=xj(t) +M∑
l=1&l 6=jnl
xj(t)− xl(t)
|xj(t)− xl(t)|2
].
Proof. For any fixed X ∈ DM , since Q is a harmonic function, there exists a function
ϕ1(x) such that
J∇xQ (x;X) = ∇ϕ1(x), x ∈ D.
Thus, ϕ1(x) satisfies the Laplace equation
∆ϕ1(x) = ∇ · (J∇xQ(x;X)) = ∂yxϕ1(x)− ∂xyϕ1(x) = 0, x ∈ D, (2.20)
with the following Neumann BC
∂νϕ1(x) = (J∇xQ(x;X)) · ν = ∇xQ(x;X) · ν⊥ = ∂ν⊥Q(x;X), x ∈ ∂D. (2.21)
Noticing (2.17), we obtain for x ∈ ∂D,
∂νϕ1(x) = ∂ν⊥ω(x)−∂
∂ν⊥
M∑
l=1
nlθ(x−xl) = ∂ν⊥ω(x)−∂
∂ν
M∑
l=1
nl ln |x−xl|. (2.22)
Combining (2.20), (2.22), (2.13) and (2.14), we get
∆(R(x;X)−ϕ1(x)) = 0, x ∈ D, ∂ν (R(x;X)− ϕ1(x)) = 0, x ∈ ∂D. (2.23)
Thus
R(x;X) = ϕ1(x) + constant, x ∈ D,
which immediately implies the first equality in (2.19).
Similarly, since H is a harmonic function, there exists a function ϕ2(x) such that
J∇xH (x;X) = ∇ϕ2(x), x ∈ D.
Thus, ϕ2(x) satisfies the Laplace equation
∆ϕ2(x) = ∇ · (J∇xH(x;X)) = ∂yxϕ2(x)− ∂xyϕ2(x) = 0, x ∈ D, (2.24)
2.2 Reduced dynamical laws 21
with the following Neumann BC
∂νϕ2(x) = (J∇xH(x;X))·ν = ∇xH(x;X)·ν⊥ = ∂ν⊥H(x;X), x ∈ ∂D. (2.25)
Combining (2.24), (2.25), (2.13), (2.14) and (2.16), we get
∆(R(x;X)−ϕ2(x)) = 0, x ∈ D, ∂ν (R(x;X)− ϕ2(x)) = 0, x ∈ ∂D. (2.26)
Thus
R(x;X) = ϕ2(x) + constant, x ∈ D,
which immediately implies the second equality in (2.19).
Under homogeneous Neumann boundary condition
For the GLSE (1.1) with initial condition (2.6) under homogeneous Neumann
BC (1.4), it has been derived formally and rigorously [48,76,95] that Wbc(X) in the
renormalized energy (2.10) admit the form:
Wbc(X) = Wnbc(X) := −M∑
j=1
njR(xj ;X), (2.27)
and by using the following identity
∇xj[W (X) +Wnbc(X)] = −2nj∇x
[R(x;X) +
M∑
l=1&l 6=jnl ln |x− xl|
]
xj
, (2.28)
we have the following simplified equivalent form for (2.8):
Lemma 2.2.3. For 1 ≤ j ≤M and t > 0, system (2.8) can be simplified as
(αI+βnjJ)d
dtxj(t) = 2nj
[∇xR (x;X) |x=xj(t) +
M∑
l=1&l 6=jnl
xj(t)− xl(t)
|xj(t)− xl(t)|2
]. (2.29)
Moreover, for any fixed X ∈ DM , by introducing function H(x, X) and Q(x, X) that
both are harmonic in x satisfying respectively the boundary condition [82,83,85,107]:
∂H(x;X)
∂ν⊥= − ∂
∂ν
M∑
l=1
nlθ(x− xl), x ∈ ∂D, (2.30)
∂Q(x;X)
∂ν= − ∂
∂ν
M∑
l=1
nlθ(x− xl), x ∈ ∂D, (2.31)
2.2 Reduced dynamical laws 22
with the function θ : R2 → [0, 2π) being defined in (2.18), we have the following
lemma for the equivalence of the reduced dynamical law (2.29) [22, 23]:
Lemma 2.2.4. For any fixed X ∈ DM , we have the following identity
J∇xQ (x;X) = ∇xR (x;X) = J∇xH (x;X) , x ∈ D, (2.32)
which immediately implies the equivalence of system (2.29) and the following two
systems: for t > 0
(αI + βnjJ)d
dtxj(t) = 2nj
[∇xH (x;X) |x=xj(t) +
M∑
l=1&l 6=jnl
xj(t)− xl(t)
|xj(t)− xl(t)|2
],
(αI + βnjJ)d
dtxj(t) = 2nj
[J∇xQ (x;X) |
x=xj(t) +M∑
l=1&l 6=jnl
xj(t)− xl(t)
|xj(t)− xl(t)|2
].
Proof. Follow the line in the proof of lemma 2.2.1 and we omit the details here for
brevity.
2.2.2 Under inhomogeneous potential
It has been shown in last section that in a homogeneous potential, vortices
in the GLE dynamics under Dirichlet BC will move according to gradient flow of
the so called renormalized energy, which is associated to the BC. However, in an
inhomogeneous potential, i.e V (x) 6≡ 1, the phenomena is quite different. Generally
speaking, vortices no longer move along the gradient flow of the renomalized energy,
they move toward the critical points of the potential V (x) instead [44, 78, 80]. And
it has been proved that they obey the following reduced dynamical law [78]:
Theorem 2.2.2. As ε → 0, for j = 1, · · · , N , the vortex center xεj(t) in the GLE
dynamics with λε = 1 under Dirichlet BC will converge to point xj(t), which satisfies:
dxj(t)
dt= −∇V (xj)
V (xj), 0 ≤ t < +∞, (2.33)
xj(t = 0) = x0j . (2.34)
2.3 Numerical methods 23
In addition, for each j, if there is a Lipschitz domain Gj ⊂ D such that
x0j ∈ Gj , min
x∈∂Gj
V (x) > V (x0j ), , j = 1, · · · ,M,
the solution in (2.33) will satisfy:
for t > 0,xj(t) 6= xl(t) if j 6= l, and xj(t) ∈ Gj .
Hence, the vortices will all be pinned together to the critical points of V (x) and
further to the minimum points if V (x) has no other critical points.
Unfortunately, to our limited knowledge, there are no existing studies that deal
with the vortex dynamics in the inhomogeneous potential in the GLE dynamics
under Neumann BC, or in the CGLE or NLSE dynamics in the limiting process
ε→ 0.
2.3 Numerical methods
In this section, we present efficient and accurate numerical methods for discretiz-
ing the GLSE (1.1) with a time dependent potential U(x, t) in either a rectangle or a
disk with initial condition (1.2) and under either Dirichlet BC (1.3) or homogeneous
Neumann BC (1.4):
(λε + iβ)∂tψε(x, t) = ∆ψε +
1
ε2(U(x, t)− |ψε|2)ψε, x ∈ D, t > 0, (2.35)
here U(x, t) = V (x) +W (x, t) with W (x, t) an external potential. The key idea in
our numerical methods are based on: (i) applying a time-splitting technique which
has been widely used for nonlinear partial differential equations [67,143] to decouple
the nonlinearity in the GLSE [20,29,149,160]; and (ii) adapt proper finite difference
and/or spectral method to discretize a gradient flow with constant coefficient [18,
22, 23].
2.3.1 Time-splitting
Let τ > 0 be the time step size, denote tn = nτ for n ≥ 0. For n = 0, 1, . . ., from
time t = tn to t = tn+1, the GLSE (1.1) is solved in two splitting steps. One first
2.3 Numerical methods 24
solves
(λε + iβ)∂tψε(x, t) =
1
ε2(U(x, t)− |ψε|2)ψε, x ∈ D, t ≥ tn, (2.36)
for the time step of length τ , followed by solving
(λε + iβ)∂tψε(x, t) = ∆ψε, x ∈ D, t ≥ tn, (2.37)
for the same time step. Equation (2.37) is discretized in the next two subsections
on a rectangle and a disk, respectively. For t ∈ [tn, tn+1], to solve equation (2.36),
we rewrite
ψε(x, t) =√ρε(x, t)eiS
ε(x,t) (2.38)
with ρε and Sε being the density and phase of ψε, respectively. From (2.36), we can
easily obtain the following ODE for ρε(x, t) = |ψε(x, t)|2:
∂tρε(x, t) = η[U(x, t)− ρε(x, t)]ρε(x, t), x ∈ D, tn ≤ t ≤ tn+1, (2.39)
where η = 2λε/ε2(λ2ε + β2). Solving equation (2.39 ), we have
ρε(x, t) =ρε(x, tn) exp[ηUn(x, t)]
1 + ηρε(x, tn)∫ ttnexp[ηUn(x, s)ds]
, (2.40)
where Un(x, t) =∫ ttnU(x, s)ds. Moreover, if W (x, t) ≡ 0, i.e., U(x, t) = V (x),
ρ(x, t) can be analytically integrated to have
ρε(x, t) =
ρε(x, tn), λε = 0,
ρε(x,tn)1+ηρε(x,tn)(t−tn) , V (x) = 0& λε 6= 0,
V (x)ρε(x,tn)ρε(x,tn)+(V (x)−ρε(x,tn)) exp[−ηV (x)(t−tn)] , V (x), λε 6= 0.
(2.41)
Plugging (2.38) back into (2.36), we obtain the equation for the phase Sε(x, t):
∂tSε(x, t) = − β
ε2(λ2ε + β2)[U(x, t)− ρε(x, t)], x ∈ D, tn ≤ t ≤ tn+1. (2.42)
Combining (2.40) and (2.42), we obtain for t ∈ [tn, tn+1],
ψε(x, t) = ψε(x, tn)√Pn(x, t) exp
[− iβ
ε2(λ2ε + β2)(Un(x, t)−
∫ t
tn
ρε(x, s)ds)
], (2.43)
2.3 Numerical methods 25
where
Pn(x, t) =exp[ηUn(x, t)]
1 + η|ψε(x, tn)|2∫ ttnexp[ηUn(x, s)]ds
. (2.44)
Furthermore, if U(x, t) = V (x), we have the explicit formation of ψε:
ψε(x, t) = ψε(x, tn)
exp[− iβε2β2 (V (x)− |ψε(x, tn)|2)(t− tn)
], λε = 0,
√P (x, t) exp
[− iβ
2λ2εln P (x, t)
], λε 6= 0
(2.45)
where
P (x, t) =
11+η|ψε(x,tn)|2(t−tn) , V (x) ≡ 0,
V (x)|ψε(x,tn)|2+(V (x)−|ψε(x,tn)|2) exp(−ηV (x)(t−tn)) , V (x) 6≡ 0,
(2.46)
Remark 2.3.1. If functions Un(x, t) and other integrals in (2.43) cannot be calcu-
lated analytically, numerical quadrature such as the trapezoidal rule can be applied
to solve them.
Remark 2.3.2. In practice, we always use the second-order Strang splitting [143],
that is, from time t = tn to t = tn+1: (i) evolve (2.36) for half time step τ/2 with
initial data given at t = tn; (ii) evolve (2.37) for one step τ starting with the new
data; and (iii) evolve (2.36) for half time step τ/2 again with the newer data.
2.3.2 Discretization in a rectangular domain
Let D = [a, b] × [c, d] be a rectangular domain, and denote mesh sizes hx=b−aN
and hy=d−cL
with N and L being two even positive integers.
First we present a Crank-Nicolson 4th-order compact finite difference (CNFD)
method for discretizing the equation (2.37) with Dirichlet BC (1.3) by using the
4th-order compact finite difference discretization for spatial derivatives followed by
a Crank-Nicolson scheme for temporal derivative. In order to do so, denote the grid
points as xj = a+jhx for j = 0, 1, . . . , N and yl = c+lhy for l = 0, 1, . . . , L; and ψε,nj,l
2.3 Numerical methods 26
be the numerical approximation of ψε(xj , yl, tn) for j = 0, 1, . . . , N , l = 0, 1, . . . , L
and n ≥ 0. Define the finite difference operators as
δ2xψε,nj,l =
ψε,nj+1,l − 2ψε,nj,l + ψε,nj−1,l
h2x, δ2yψ
ε,nj,l =
ψε,nj,l+1 − 2ψε,nj,l + ψε,nj,l−1
h2y,
then a CNFD discretization for (2.37) reads, i.e., for 1 ≤ j ≤ N−1 and 1 ≤ l ≤ L−1
(λε + iβ)
τ
[I +
h2x12δ2x +
h2y
12δ2y
](ψε,n+1j,l − ψ
ε,nj,l
)=
[δ2x + δ2y +
h2x + h2y
12δ2xδ
2y
](ψε,n+1j,l + ψ
ε,nj,l
2
),
(2.47)
where I is the identity operator and the boundary condition (1.3) is discretized as
ψε,n+10,l = g(a, yl), ψε,n+1
M,l = g(b, yl), l = 0, 1, . . . , L,
ψε,n+1j,0 = g(xj, c), ψε,n+1
j,L = g(xj, d), j = 0, 1, . . . , N.
Here although an implicit time discretization is applied for (2.37), the linear system
in (2.47) can be solved explicitly via direct Poisson solver through DST [93] at the
computational cost of O (NL ln(NL)).
Combining the above CNFD discretization with the second order Strang splitting
presented in the previous subsection, we obtain a time-splitting Crank-Nicolson finite
difference (TSCNFD) discretization for the GLSE (1.1) on a rectangle with Dirichlet
BC (1.3). This TSCNFD discretization is unconditionally stable, second order in
time and fourth order in space, the memory cost is O(NL) and the computational
cost per time step is O (NL ln(NL)).
Next we present a cosine pseudospectral method for the equation (2.37) with
homogeneous Neumann BC (1.4) by using cosine spectral discretization for spatial
derivatives followed by integrating in time exactly. To this end, let
YNL = spanφpq(x) = cos(µxp(x−a)) cos(µyq(y−c)), 0 ≤ p ≤ N−1, 0 ≤ q ≤ L−1,
with
µxp =pπ
b− a, p = 0, 1, . . . , N − 1; µyq =
qπ
d− c, q = 0, 1, . . . , L− 1.
2.3 Numerical methods 27
Then the cosine spectral discretization for (2.37) with (1.4) is as follows:
Find ψεNL(x, t) ∈ YNL, i.e.,
ψεNL(x, t) =N−1∑
p=0
L−1∑
q=0
ψεpq(t)φpq(x), x ∈ D, t ≥ tn, (2.48)
such that
(λε + iβ)∂tψεNL(x, t) = ∆ψεNL(x, t), x ∈ D, t ≥ tn. (2.49)
Plugging (2.48) into (2.49), noticing the orthogonality of the cosine functions, for
0 ≤ p ≤ N − 1 and 0 ≤ q ≤ L− 1, we find
(λε + iβ)d
dtψεpq(t) = −
[(µxp)
2 + (µyq)2]ψεpq(t), t ≥ tn. (2.50)
The above ODE can be integrated exactly in time, i.e.,
ψεpq(t) = eη[(µxp)
2+(µyq )2](t−tn) ψεpq(tn), t ≥ tn, 0 ≤ p ≤ N−1, 0 ≤ q ≤ L−1, (2.51)
where η = iβ−λελ2ε+β
2 . The above procedure is not suitable in practice due to the difficulty
of computing the integrals in (2.48). In practice, we need approximate the integrals
by a quadrature rule on grids. Define the grid points as xj+ 12= a + (j + 1
2)hx
for j = 0, 1, . . . , N − 1 and yl+ 12= c + (l + 1
2)hy for j = 0, 1, . . . , L − 1; denote
ψε,nj+ 1
2,l+ 1
2
be the numerical approximation of ψε(xj+ 12, yl+ 1
2, tn) for j = 0, 1, . . . , N−1,
l = 0, 1, . . . , L − 1 and n ≥ 0; and ψε,n be the solution vector at time t = tn
with components ψε,nj+ 1
2,l+ 1
2
, 0 ≤ j ≤ N − 1, 0 ≤ l ≤ L − 1 for n ≥ 0. Choose
ψε,0j+ 1
2,l+ 1
2
= ψε0(xj+ 12, yl+ 1
2) for 0 ≤ j ≤ N − 1 and 0 ≤ l ≤ L − 1, then a cosine
pseudospectral approximation for (2.37) with (1.4) reads as, for 0 ≤ j ≤ N − 1 and
0 ≤ l ≤ L− 1
ψε,n+1
j+ 12,l+ 1
2
=N−1∑
p=0
L−1∑
q=0
αxpαyqeη[(µxp)2+(µyq )
2]τ ψε,npq φpq(xj+ 12, yl+ 1
2), n ≥ 0, (2.52)
where
ψε,np,q = αxpαyq
N−1∑
j=0
L−1∑
l=0
ψε,nj+ 1
2,l+ 1
2
φpq(xj+ 12, yl+ 1
2), 0 ≤ p ≤ N − 1, 0 ≤ q ≤ L− 1,
2.3 Numerical methods 28
with
αxp =
√1N, p = 0,√
2N, 1 ≤ p ≤ N − 1,
αyq =
√1L, q = 0,√
2L, 1 ≤ q ≤ L− 1.
Again, combining the above cosine pseudospectral discretization with the second
order Strang splitting presented in the subsection 3.1, we obtain a time-splitting co-
sine pseudospectral (TSCP) discretization for the GLSE (1.1) on a rectangle with
homogeneous Neumann BC (1.4). This TSCP discretization is unconditionally sta-
ble, second order in time and spectral order in space, the memory cost is O(NL)
and the computational cost per time step is O (NL ln(NL)) via DCT [140].
Remark 2.3.1. If the homogeneous Neumann BC (1.4) is replaced by periodic BC,
the above TSCP discretization for the GLSE (1.1) is still valid provided that we
replace the cosine basis functions by the Fourier basis functions in the spectral dis-
cretization and use the quadrature rule associated to the Fourier functions [140]. We
omit the details here for brevity.
2.3.3 Discretization in a disk domain
Let D = x | |x| < R be a disk with R > 0 a fixed constant. In this case,
it is natural to adopt the polar coordinate (r, θ). In order to discretize (2.37) with
either (1.3) or (1.4), we apply the standard Fourier pseudospectral method in θ-
direction [140], finite element method in r-direction, and Crank-Nicolson method in
time [13, 18, 160]. With the following truncated Fourier expansion
ψε(r, θ, t) =
l=L/2−1∑
l=−L/2ψl(r, t)e
ilθ, 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, (2.53)
where L is an even positive number and ψl is the Fourier coefficients for the l-
th mode, plugging (2.53) into (2.37) and using the orthogonality of the Fourier
functions, we obtain for l=−L2, · · · , L
2− 1:
(λε+ iβ)∂tψl(r, t) =1
r∂r
(r∂rψl(r, t)
)− l2
r2ψl(r, t), 0 < r < R, t ≥ tn, (2.54)
2.3 Numerical methods 29
with the following boundary condition at r = 0
∂rψ0(0, t) = 0, ψl(0, t) = 0, l 6= 0, t ≥ tn. (2.55)
When the Dirichlet BC (1.3) is used for (2.37), we then impose the following
boundary condition at r = R:
ψl(R, t) = gl :=1
2π
∫ 2π
0
g(θ)e−ilθdθ, −L2≤ l ≤ L
2− 1, t ≥ tn. (2.56)
Let P k denote all polynomials with degree at most k, denote 0 = r0 < r1 < · · · <rN = R be a partition for the interval [0, R] with N a positive integer and a mesh
size h = max0≤j≤N−1(rj+1 − rj), and define a finite element space by
Uh =uh ∈ C[0, R] | uh|[rj,rj+1] ∈ P k, 0 ≤ j ≤ N − 1
.
Introducing the following finite element approximate sets associated to the Dirichlet
BCs for −L2≤ l ≤ L
2− 1 as
Ugl =
uh ∈ Uh | uh(R) = g0
, l = 0,
uh ∈ Uh | uh(0) = 0, uh(R) = gl
, l 6= 0;
(2.57)
then we obtain the FEM approximation for (2.54) with (2.55) and (2.56):
Find ψhl (·, t) ∈ Ugl with −L
2≤ l ≤ L
2− 1 such that
d
dtA(ψhl , φ
h) = B(ψhl , φh) + l2C(ψhl , φ
h), ∀φh ∈ U0l , tn ≤ t ≤ tn+1; (2.58)
where the bilinear forms A, B and C are defined as
A(uh, vh) = (λε + iβ)
∫ R
0
ruhvhdr, B(uh, vh) = −∫ R
0
r∂ruh∂rv
hdr,
C(uh, vh) = −∫ R
0
1
ruhvhdr, ∀uh, vh ∈ Uh.
The above ODE system (2.58) is then discretized by the standard Crank-Nicolson
scheme in time. Here although an implicit time discretization is applied for (2.58),
the one-dimensional nature of the problem makes the coefficients matrix for the
linear system band limited. For example, if the piecewise linear polynomial is used,
2.3 Numerical methods 30
i.e., k = 1 in Uh, the matrix is tridiagonal. Thus for each fixed −L2≤ l ≤ L
2−1, fast
algorithms can be applied to solve the resulting linear systems at the cost of O(N).
Similarly, when the homogeneous Neumann BC (1.4) is used for (2.37), the above
discretization is still valid provided that we replace the boundary condition at r = R
in (2.56) by
∂rψl(R, t) = 0, −L2≤ l ≤ L
2− 1, t ≥ tn, (2.59)
the finite element subsets Ugl in (2.57) and U0
l in (2.58) by the following finite element
spaces
Unl =
Uh, l = 0,uh ∈ Uh | uh(0) = 0
, l 6= 0.
(2.60)
The detailed discretization is omitted here for brevity.
Remark 2.3.2. The equation (2.54) with (2.55) can also be discretized in space
by either Legendre or Chebyshev pseudospectral method [140] and in time by the
Crank-Nicolson method.
Chapter 3
Vortex dynamics in GLE
Formal analysis indicate that, if initially ψε0 has isolated vortices, these vortices
move with velocities of the order of | ln ε|−1 in the GLE dynamics with λε = 1
[28, 99, 102]. Therefore, to obtain nontrivial vortex dynamics, in this chapter, we
always assume 0 < ε < 1 and choose
λε =1
| ln ε| =1
ln(1/ε), 0 < ε < 1. (3.1)
We then apply the numerical method presented in chapter 2 to simulate quan-
tized vortex interaction of GLE, i.e., β = 0, λε = 1ln(1/ε)
in the GLSE (1.1), with
different ε and under different initial setups including single vortex, vortex pair,
vortex dipole and vortex cluster. We study how the dimensionless parameter ε, ini-
tial setup, boundary value and geometry of the domain D affect the dynamics and
interaction of vortices. Moreover, we compare the results obtained from the GLE
with those from the corresponding reduced dynamical laws, and identify the cases
where the reduced dynamical laws agree qualitatively and/or quantitatively as well
as fail to agree with those from GLE on vortex interaction. Finally, we also obtain
numerically different patterns of the steady states for quantized vortex clusters and
study the alignment of the vortices in the steady state.
31
3.1 Initial setup 32
3.1 Initial setup
For a given bounded domain D, the GLSE (1.1) is unchanged by the re-scaling
x → lx, t→ l2t and ε → lε with l the diameter of D. Thus without lose of generality,
hereafter, without specification, we always assume that the diameter of D is O(1).
The initial data ψε0 in (1.2) for simulating GLE thus is chosen as (2.6) with R0 = 0.25
in (2.7). To simulate GLE under Dirichlet BC, we choose the function g(x) in (1.3)
as
g(x) = ei(h(x)+∑M
j=1 njθ(x−x0j )), x ∈ ∂D, (3.2)
and we consider following six kinds of modes for the phase shift h(x) in (3.2) and
(2.6):
• Mode 0: h(x) = 0, Mode 1: h(x) = x+ y,
• Mode 2: h(x) = x− y, Mode 3: h(x) = x2 − y2,
• Mode 4: h(x) = x2 − y2 − 2xy, Mode 5: h(x) = x2 − y2 − 2xy.
Moreover, to simulate GLE under homogeneous Neumann BC, we choose the phase
shift h(x) to be the solution of the following problem:
∆h(x) = 0, x ∈ D,∂∂νh(x) = − ∂
∂ν
∑Ml=1 nlθ(x− xl), x ∈ ∂D,
∫D h(x)dx = 0.
(3.3)
Without specification, this initial setup will also be used in chapter 4 in studying
vortex interaction in the NLSE dynamics and chapter 5 in the CGLE dynamics. To
simplify our presentation, for j = 1, 2, . . . ,M , hereafter we let xεj(t) and xrj(t) be
the j-th vortex center in the GLE dynamics and corresponding reduced dynamics,
respectively, and denote dεj(t) = |xεj(t)− xrj(t)| as their difference. Moreover, in the
presentation of figures, the initial location of a vortex with winding number +1, −1
and the location that two vortices merge are marked as ‘+’, ‘’ and ‘⋄’, respectively.
3.2 Numerical results under Dirichlet BC 33
Furthermore, if a vortex could finally stay steady in somewhere, we denote its final
location as ‘*’. Finally, in our computations, if not specified, we take D = [−1, 1]2,
mesh sizes hx = hy = ε10
and time step τ = 10−6. The GLE with (1.3), (1.2) and
(2.6) is solved by the method TSCNFD presented in section 3.
3.2 Numerical results under Dirichlet BC
3.2.1 Single vortex
Here we present numerical results of the motion of a single quantized vortex
under the GLE dynamics and its corresponding reduced dynamical laws. We take
M = 1, n1 = 1 and consider following cases: case I. x01 = (0, 0), h(x) = x + y;
case II. x01 = (0, 0), h(x) = x − y; case III. x0
1 = (0, 0), h(x) = x2 − y2; case IV.
x01 = (0.1, 0.2), h(x) = x + y; case V. x0
1 = (0.1, 0.2), h(x) = x − y; and case VI.
x01 = (0.1, 0.2), h(x) = x2−y2. Fig. 3.1 depicts trajectory of the vortex center when
ε = 132
for the above 6 cases and dε1 with different ε for case II, IV and VI. From Fig.
3.1 and additional numerical experiments not shown here for brevity, we can draw
following conclusions: (i). When h(x) ≡ 0, the vortex center doesn’t move and this
is similar to the case in the whole space. (ii). When h(x) = (x+by)(x− yb) with b 6= 0,
the vortex does not move if x0 = (0, 0), while it does move if x0 6= (0, 0) (cf. case
III and VI for b = 1). (iii). When h(x) 6= 0 and h(x) 6= (x+ by)(x− yb) with b 6= 0,
in general, the vortex center does move to a different point from its initial location
and stays there forever. This is quite different from the situation in the whole space,
where a single vortex may move to infinity under the initial data (2.6) with h(x) 6= 0
and D = R2. (iv). In general, the initial location, the geometry of the domain and
the boundary value will all affect the motion of the vortex center. (v). When ε→ 0,
the dynamics of the vortex center in the GLE dynamics converges uniformly in
time to that in the reduced dynamics (cf. Fig. 3.1) which verifies numerically the
validation of the reduced dynamical laws. In fact, based on our extensive numerical
experiments, the motion of the vortex center from the reduced dynamical laws agree
3.2 Numerical results under Dirichlet BC 34
(a)-1 0 1
-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
(b)-1 0 1
-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
(c)0 0.3 0.6
0
0.1
0.2
t
d 1ε
0 0.61 1.220
0.07
0.14
t
d 1ε
0 0.75 1.50
0.21
0.42
t
d 1ε
ε=1/25
ε=1/50
ε=1/100
ε=1/25
ε=1/50
ε=1/100
ε=1/25
ε=1/50
ε=1/100
Figure 3.1: (a)-(b): Trajectory of the vortex center in GLE under Dirichlet BC when
ε = 132
for cases I-VI (from left to right and then from top to bottom), and (c): dε1
for different ε for cases II, IV and VI (from left to right) in section 3.2.1.
with those from the GLE dynamics qualitatively when 0 < ε < 1 and quantitatively
when 0 < ε≪ 1.
3.2.2 Vortex pair
Here we present numerical results of the interaction of vortex pair under the
GLE dynamics and its corresponding reduced dynamical laws, i.e., we take M = 2,
n1 = n2 = 1, x01 = (−0.5, 0) and x0
2 = (0.5, 0) in (2.6). Fig. 3.2 depicts time
evolution of the amplitude |ψε|, while Fig. 3.3 shows that of the GL functionals as
3.2 Numerical results under Dirichlet BC 35
well as the trajectory of the vortex centers when ε = 132
in with different h(x) in
(2.6). Fig. 3.4 shows time evolution of xr1(t), xε1(t) and d
ε1(t) with different h(x) in
(2.6).
From Figs. 3.2, 3.3 & 3.4 and additional numerical results not shown here for
brevity, we can draw the following conclusions for the interaction of vortex pair un-
der the GLE dynamics with Dirichlet BC: (i). The two vortices undergo a repulsive
interaction, they never collide, both of them move towards the boundary of D for a
while and finally stop somewhere near the boundary which indicate that the bound-
ary imposes a repulsive force on the vortices when t is large enough (cf. Figs. 3.2
& 3.3). (ii). When h(x) ≡ 0, the two vortex centers move outward along the line
connecting them initially, and their trajectories are symmetric, i.e., xε1(t) = −xε2(t),
while when h(x) 6= 0, it affects the motion of the two vortex centers significantly
(cf. Fig. 3.3). (iii). When ε → 0, the dynamics of the two vortex centers in the
GLE dynamics converges uniformly in time to that in the reduced dynamics (cf.
Fig. 3.4) which verifies numerically the validation of the reduced dynamical laws
in this case. In fact, based on our extensive numerical experiments, the motions of
the two vortex centers from the reduced dynamical laws agree with those from the
GLE dynamics qualitatively when 0 < ε < 1 and quantitatively when 0 < ε ≪ 1.
(iv). During the dynamics of GLE, the GL functional and its kinetic part decrease
when time increases, its interaction part changes dramatically when t is small, and
when t → ∞, all the three quantities converge to constants (cf. Fig. 3.3), which
immediately imply that a steady state solution will be reached when t→ ∞.
3.2.3 Vortex dipole
Here we present numerical results of the interaction of vortex dipole under the
GLE dynamics and its corresponding reduced dynamical laws, i.e., we take M = 2,
n1 = −1, n2 = 1, x01 = (−d0, 0) and x0
2 = (d0, 0) in (2.6). Fig. 3.5 depicts time
evolution of the amplitude |ψε|, while Fig. 3.6 shows that of the GL functionals as
well as the trajectory of the vortex centers when ε = 132
in GLE with different d0
3.2 Numerical results under Dirichlet BC 36
(a)
(b)
Figure 3.2: Contour plots of |ψε(x, t)| at different times for the interaction of vortex
pair in GLE under Dirichlet BC with ε = 132
and different h(x) in (2.6): (a) h(x) = 0,
(b) h(x) = x+ y.
(a) -0.8 0 0.8-0.8
0
0.8
x
y
0 0.2 0.420
23
26
t
0 0.2 0.43.123
3.139
3.155
t
E ε
E ε
kin
E ε
int
(b) -0.6 0.15 0.9-0.9
-0.15
0.6
x
y
0 1.5 318
24
30
t
0 1.5 33.11
3.16
3.21
t
E ε
E ε
kin
E ε
int
Figure 3.3: Trajectory of vortex centers (left) and time evolution of the GL func-
tionals (right) for the interaction of vortex pair in GLE under Dirichlet BC with
ε = 132
for different h(x) in (2.6): (a) h(x) = 0, (b) h(x) = x+ y.
and h(x) in (2.6) as well as the critical value dεc for different ε when h(x) ≡ x + y.
Fig. 3.7 shows time evolution of xr1(t), xε1(t) and d
ε1(t) with d0 = 0.5 for different ε
and h(x) in (2.6).
From Figs. 3.5, 3.6 & 3.7 and additional numerical results not shown here for
3.2 Numerical results under Dirichlet BC 37
(a) 0 0.075 0.15-0.1
0
0.1
t
y 1 ε(t
) or
y1 r(t
)
ε=1/32
ε=1/128
reduced
0 0.2 0.4-0.78
-0.64
-0.5
t
x 1 ε(t
) or
x 1 r
(t)
ε=1/16
ε=1/128
reduced
0 0.2 0.4
0.022
0.045
d 1ε
t
ε=1/32
ε=1/64
ε=1/128
(b) 0 1 2-0.5
-0.35
-0.15
t
x 1 ε(t
) or
x 1 r
(t)
ε=1/16
ε=1/64
reduced
0 1 2-0.8
-0.4
0
t
y 1 ε(t
) or
y 1 r
(t)
ε=1/16
ε=1/64
reduced
0 1 2
0.08
0.16
d 1ε
t
ε=1/16
ε=1/32
ε=1/64
Figure 3.4: Time evolution of xε1(t) and xr1(t) (left and middle) and their difference
dε1 (right) for different ε for the interaction of vortex pair in GLE under Dirichlet
BC for different h(x) in (2.6): (a) h(x) = 0, (b) h(x) = x+ y.
brevity, we can draw the following conclusions for the interaction of vortex dipole
under the GLE dynamics with Dirichlet BC: (i). Both boundary value, i.e., h(x),
and distance between the two vortex centers initially, i.e., 2d0, affect the motion of
the vortices significantly. (ii). When h(x) ≡ 0, for any initial location of the vortex
dipole, the two vortices always undergo an attractive interaction and their centers
move toward each other along the line connecting them initially, their trajectory are
symmetric with respect to the line perpendicular to the segment connecting them
initially, and finally, they merge at the middle point of this segment, i.e., the point
xmerge =12(x0
1+x02) (cf. Figs. 3.5 & 3.6). At the collision, both vortices in the vortex
dipole merge/annihilate with each other; and after the collision, they will disappear
and no vortex is left afterwards during the dynamics. For any fixed 0 < ε < 1, there
is a collision time Tε which increases when ε decreases. (iii). When h(x) = x + y,
the two vortices move along a symmetric trajectory, i.e., xε1(t) = −xε2(t). Moreover,
for the reduced dynamical laws, there exists a critical value drc, which is found
3.2 Numerical results under Dirichlet BC 38
(a)
(b)
(c)
Figure 3.5: Contour plots of |ψε(x, t)| at different times for the interaction of vortex
dipole in GLE under Dirichlet BC with ε = 132
for different d0 and h(x) in (2.6): (a)
h(x) = 0, d0 = 0.5, (b) h(x) = x+ y, d0 = 0.5, (c) h(x) = x+ y, d0 = 0.3.
numerically as drc ≈ 0.4142, such that when d0 < drc, then the vortex dipole will
merge at finite time, and respectively, when d0 > drc, the vortex dipole will never
collide. Similarly, for the vortex dipole under the GLE dynamics, for each fixed
0 < ε < 1, there exists a critical value dεc such that when d0 < dεc, then the vortex
dipole will merge at finite time, and respectively, when d0 > dεc, the vortex dipole
will never collide (cf. Figs. 3.5 & 3.6). We find numerically the critical distance
dεc for 0 < ε < 1 and depict them in Fig. 3.8. From these values, we can fit the
3.2 Numerical results under Dirichlet BC 39
(a) -0.5 0 0.5
-0.5
0
0.5
x
y
0 0.07 0.140
12
24
t
0 0.07 0.140
2
4
t
E ε
E ε
kin
E ε
int
(c) -0.4 0 0.4-0.1
0
0.1
x
y
0 0.09 0.185
14.5
24
t
0 0.09 0.180
1.9
3.8
t
E ε
E ε
kin
E ε
int
(b) -0.6 0 0.6-0.6
0
0.6
x
y
0 0.75 1.518
21
24
t
0 0.75 1.53.12
3.14
3.16
t
E ε
E ε
kin
E ε
int
(d) 0.04 0.08 0.120.414
0.423
0.432
εd
cε-3.1 -2.6 -2.1
-7.7
-5.8
-4.0
ln(ε)
ln(d
cε -dcr )
Figure 3.6: (a)-(c): Trajectory of vortex centers (left) and time evolution of the
GL functionals (right) for the interaction of vortex dipole in GLE under Dirichlet
BC with ε = 132
for different d0 and h(x) in (2.6): (a) h(x) = 0, d0 = 0.5, (b)
h(x) = x+y, d0 = 0.5, (c) h(x) = x+y, d0 = 0.3. (d): Critical value dεc for different
ε when h(x) ≡ x+ y.
following relationship between dεc and drc:
dεc ≈ drc + 41.26ε3.8, 0 ≤ ε < 1.
(iv). When ε → 0, the dynamics of the two vortex centers under the GLE dynamics
converges uniformly in time to that of the reduced dynamical laws before the collision
happens (cf. Fig. 3.7) which verifies numerically the validation of the reduced
dynamical laws in this case. In fact, based on our extensive numerical experiments,
the motion of the two vortex centers from the reduced dynamical laws agree with
those from the GLE dynamics qualitatively when 0 < ε < 1 and quantitatively
when 0 < ε ≪ 1 if the initial distance between the two vortex centers satisfies either
0 < d0 < drc or d0 > dεc. On the contrary, if drc < d0 < dεc, then the motion of
the vortex dipole from the reduced dynamical laws is different qualitatively from
that of the GLE dynamics. (v). During the dynamics of GLE, the GL functional
decreases when time increases, its kinetic and interaction parts change dramatically
3.2 Numerical results under Dirichlet BC 40
(a) 0 0.075 0.15-0.1
0
0.1
t
y 1 ε(t
) or
y1 r(t
)
ε=1/32
ε=1/128
reduced
0 0.07 0.14
0.08
0.16
d 1ε
t
ε=1/64
ε=1/128
ε=1/200
0 0.075 0.15-0.5
-0.25
0
t
x 1 ε(t
) or
x 1 r
(t)
ε=1/32
ε=1/128
reduced
(b) 0 0.75 1.5-0.28
-0.14
0
t
y 1 ε(t
) or
y1 r (t
)
ε=1/16
ε=1/64
reduced
0 0.75 1.50
0.03
0.06
d 1ε
t
ε=1/16
ε=1/32
ε=1/64
0 0.75 1.5
-0.58
-0.54
-0.5
t
x 1 ε(t
) or
x 1 r
(t)
ε=1/16
ε=1/64
reduced
Figure 3.7: Time evolution of xε1(t), xr1(t) (left and middle) and their difference dε1
(right) for different ε for the interaction of vortex dipole in GLE under Dirichlet BC
with d0 = 0.5 for different h(x) in (2.6): (a) h(x) = 0, (b) h(x) = x+ y.
0.04 0.08 0.120.414
0.423
0.432
ε
d cε
-3.1 -2.6 -2.1-7.7
-5.8
-4.0
ln(ε)
ln(d
cε -dcr )
Figure 3.8: Critical value dεc for the interaction of vortex dipole of the GLE under
Dirichlet BC with h(x) ≡ x+ y in (2.6) for different ε.
when t is small, and when t→ ∞, all the three quantities converge to constants (cf.
Fig. 3.6). Moreover, if finite time merging/annihilation happens, the GL functional
and its kinetic and interaction parts change significantly during the collision. In
addition, when t → ∞, the interaction energy goes to 0 which immediately implies
that a steady state will be reached in the form of φε(x) = eic(x), where c(x) is a
3.2 Numerical results under Dirichlet BC 41
harmonic function satisfying c(x)|∂D = h(x) +∑M
j=1 njθ(x− x0j ).
3.2.4 Vortex cluster
Here we present numerical results of the interaction of vortex clusters under
the GLE dynamics. We will consider the following cases: case I. M = 3, n1 =
n2 = n3 = 1, x01 = (−0.25,
√34), x0
2 = (−0.25,−√34), x0
3 = (0.5, 0); case II. M = 3,
n1 = n2 = n3 = 1, x01 = (−0.4, 0), x0
2 = (0, 0), x03 = (0.4, 0); case III. M = 3,
n1 = n2 = n3 = 1, x01 = (0, 0.3), x0
2 = (0.15, 0.15), x03 = (0.3, 0); case IV. M = 3,
n1 = −1, n2 = n3 = 1, x01 = (−0.25,
√34), x0
2 = (−0.25,−√34), x0
3 = (0.5, 0); case V.
M = 3, n2 = −1, n1 = n3 = 1, x01 = (−0.4, 0), x0
2 = (0, 0), x03 = (0.4, 0); case VI.
M = 3, n1 = −1, n2 = n3 = 1, x01 = (0.2, 0.3), x0
2 = (−0.3, 0.4), x03 = (−0.4,−0.2);
case VII. M = 4, n1 = n2 = n3 = n4 = 1, x01 = (0, 0.5), x0
2 = (−0.5, 0), x03 =
(0,−0.5), x04 = (0.5, 0); case VIII. M = 4, n1 = n3 = −1, n2 = n4 = 1, x0
1 = (0, 0.5),
x02 = (−0.5, 0), x0
3 = (0,−0.5), x04 = (0.5, 0); and case IX. M = 4, n1 = n2 = −1,
n3 = n4 = 1, x01 = (0, 0.5), x0
2 = (−0.5, 0), x03 = (0,−0.5), x0
4 = (0.5, 0). Fig.
3.9 shows trajectory of the vortex centers when ε = 132
in and h(x) = 0 in (2.6)
for the above 9 cases. From Fig. 3.9 and additional numerical experiments not
shown here for brevity, we can draw the following conclusions: (i). The interaction
of vortex clusters under the GLE dynamics with Dirichlet BC is very interesting
and complicated. The detailed dynamics and interaction pattern of a vortex cluster
depends on its initial alignment of the vortices, geometry of the domain D and the
boundary value g(x). (ii). For a cluster of M vortices, if they have the same index,
then no collision will happen for any time t ≥ 0. On the other hand, if they have
opposite index, e.g. M+ > 0 vortices with index ‘+1’ and M− > 0 vortices with
index ‘−1’ satisfying M+ +M− = M , collision will always happen at finite time.
In addition, when t is sufficiently large, there exist exactly |M+ −M−| vortices of
winding number ‘+1’ ifM+ > M−, and resp. ‘−1’ ifM+ < M−, left in the domain.
3.2 Numerical results under Dirichlet BC 42
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-0.45 0.1 0.65-0.6
0
0.6
x
y
-0.55 0 0.55-0.6
0
0.6
x
y
-0.65 -0.2 0.25-0.4
0.1
0.6
x
y
-0.9 0 0.9-0.9
0
0.9
x
y
-0.55 0 0.55-0.6
0
0.6
x
y
-0.55 0 0.55-0.6
0
0.6
x
y
Figure 3.9: Trajectory of vortex centers for the interaction of different vortex
clusters in GLE under Dirichlet BC with ε = 132
and h(x) = 0 for cases I-IX (from
left to right and then from top to bottom) in section 3.2.4.
3.2.5 Steady state patterns of vortex clusters
Here we present the steady state patterns of vortex clusters in the GLE dynamics
under Dirichlet BC. We study how the geometry of the domain D and boundary
condition affect the alignment of vortices in the steady states. To this end, we take
ε = 116
in,
nj = 1, x0j = 0.5
(cos
(2jπ
M
), sin
(2jπ
M
)), j = 1, 2, . . . ,M,
i.e., initially we have M like vortices which are located uniformly in a circle centered
at origin with radius R1 = 0.5.
3.2 Numerical results under Dirichlet BC 43
(a)
(b)
(c)
Figure 3.10: Contour plots of |φε(x)| for the steady states of vortex cluster in GLE
under Dirichlet BC with ε = 116
for M = 8, 12, 16, 20 (from left column to
right column) and different domains: (a) unit disk D = B1(0), (b) square domain
D = [−1, 1]2, (c) rectangular domain D = [−1.6, 1.6]× [−1, 1].
Denote φε(x) as the steady state, i.e., φε(x) = limt→∞ ψε(x, t) for x ∈ D. Fig.
3.10 depicts the contour plots of the amplitude |φε| of the steady state in the GLE
dynamics with h(x) = x2 − y2 + 2xy in (2.6) for different M and domains, while
Fig. 3.11 depicts similar results on a rectangular domain D = [−1.6, 1.6] × [−1, 1]
for different M and h(x) in (2.6). In addition, Fig. 3.11 shows similar results with
M = 8 for different h(x) and domain D.
From Figs. 3.10, 3.11 & 3.12 and additional numerical results not shown here
for brevity, we can draw the following conclusions for the steady state patterns
of vortex clusters under the GLE dynamics with Dirichlet BC: (i). The vortex
undergo repulsive interaction between each other and they move to locations near
the boundary of D, there is no collision and a steady state pattern is formed when
3.2 Numerical results under Dirichlet BC 44
(a)
(b)
(c)
(d)
(e)
Figure 3.11: Contour plots of |φε(x)| for the steady states of vortex cluster in GLE
under Dirichlet BC with ε = 116
on a rectangular domain D = [−1.6, 1.6] × [−1, 1]
for M = 8, 12, 16, 20 (from left column to right column) and different h(x): (a)
h(x) = 0, (b) h(x) = x + y, (c) h(x) = x2 − y2, (d) h(x) = x − y, (e) h(x) =
x2 − y2 − 2xy.
t → ∞. In fact, the steady state is also the solution of the following minimization
problem
φε = argminφ(x)|x∈∂D=ψε
0(x)|x∈∂DEε(φ).
Actually, based on our extensive numerical experiments, we found that for a vortex
cluster of any configuration, i.e., vortices in the vortex cluster may be opposite
winding number, the vortices either merge and annihilate and all the leftover vortices
3.2 Numerical results under Dirichlet BC 45
Figure 3.12: Contour plots of |φε(x)| for the steady states of vortex cluster in GLE
under Dirichlet BC with ε = 116
and M = 8 on a unit disk D = B1(0) (top row) or
a square D = [−1, 1]2 (bottom row) under different h(x) = 0, x + y, x2 − y2, x −y, x2 − y2 − 2xy (from left column to right column).
(a)2 8 14
0.05
0.15
0.25
M
LW
0.7 1.7 2.7
-3
-2.2
-1.4
ln(M)
ln(L
W)
(b)2 8 14
0.05
0.15
0.25
M
LW
0.7 1.7 2.7
-3
-2.2
-1.4
ln(M)
ln(L
W)
Figure 3.13: Width of the boundary layer LW vs M (the number of vortices) under
Dirichlet BC on a square D = [−1, 1]2 when ε = 116
for different h(x): (a) h(x) = 0,
(b) h(x) = x+ y.
are all pinned in near the boundary finally. This phenomena is similar with the one
in the superconductor involving magnetic field [104].
(ii). During the dynamics, the GL functional decreases when time increases.
(iii). Both the geometry of the domain and the boundary condition, i.e., h(x), affect
the final steady states significantly. The configuration of a vortex cluster at the
steady state follows the symmetry of D and h(x). For example, in the disk domain,
when h(x) = x2 − y2 + 2xy, the vortex cluster is symmetric with respect to the two
3.2 Numerical results under Dirichlet BC 46
lines y = (1−√2)x and y = (1 +
√2)x which satisfy h(x) = 0 (cf. Fig. 3.10). (iv).
At the steady state, the distance between the vortex centers and ∂D depends on ε
and M . For fixed M , when ε decreases, the distance decreases; and respectively,
for fixed ε, when M increases, the distance decreases. In order to characterize this
distance, we denote
LW := LW (M, ε) = limt→∞
min1≤j≤M
dist(xεj(t), ∂D), M ≥ 2.
For a square domain D = [−1, 1]2, we find these distances numerically and depict
LW (M, ε) with ε = 116
for different M in Fig. 3.13. From these results, we can fit
the following relations between LW (M, ε = 1/16) as a function of M as
LW ≈ 0.4M−0.7713, M ≫ 1,
for h(x) ≡ 0, and respectively,
LW ≈ 0.3714M−0.7164, M ≫ 1,
for h(x) = x + y. For other cases, we can also fit out similar results, we omit here
for brevity.
3.2.6 Validity of RDL under small perturbation
It is well known that in the NLSE dynamics highly co-rotating vortex will radiate
out sound waves, which will in turn modify their motion. In other words, the reduced
dynamical law which does not take the radiation into account will become invalid
after the sound wave coming up, or equivalent to say, a small perturbation comes up
in the field. To understand if there were same bad situations in the GLE dynamics,
i.e., if the RDL in the GLE dynamics is valid under small perturbation or not, we
take the initial data (1.2) as
ψε(x, 0) = ψδ,ε0 (x) = ψε0(x) + δe−20((|x|−0.48)2+y2), x = (x, y) ∈ D, (3.4)
where ψε0 is given in (2.6) with h(x) ≡ 0, M = 2, n1 = n2 = 1 and x01 = −x0
2 =
(0.5, 0), i.e., we perturb the initial data for studying the interaction of a vortex pair
3.3 Numerical results under Neumann BC 47
0 0.25 0.50
0.028
0.056
t
d1δ,
ε
0 0.25 0.50
0.027
0.054
t
d1δ,
ε
δ=0,ε=1/16
δ=0,ε=1/32
δ=0,ε=1/64
δ=ε=1/16δ=ε=1/32δ=ε=1/64
Figure 3.14: Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 3.2.6
by a Gaussian function with amplitude δ. Then we take δ = ε and let ε go to
0, and solve the GLE with initial condition (3.4) for the vortex centers xδ,ε1 (t) and
xδ,ε2 (t) and compare them with those from the reduced dynamical law. We denote
dδ,εj (t) = |xδ,εj (t) − xrj(t)| for j = 1, 2 as the error. Fig. 3.14 depicts time evolution
of dδ,ε1 (t) for the case when δ = ε, i.e., small perturbation, and the case when
δ = 0, i.e., no perturbation. From this figure, we can see that small perturbation
in the initial data does not affect the motion of the vortices much, same as that
with non-perturbed initial setups, the dynamics of the two vortex centers under the
GLE dynamics with perturbed initial setups converge to those obtained from the
reduced dynamical law when ε→ 0 as well. Actually, from our extensive numerical
examples, we see that any kind of small perturbations that we consider in the initial
setup does not affect the motion of the vortex dynamics much, the RDL always hold
valid.
3.3 Numerical results under Neumann BC
3.3.1 Single vortex
Here we present numerical results of the motion of a single quantized vortex
under the GLE dynamics and its corresponding reduced dynamical laws, i.e., we
take M = 1 and n1 = 1 in (2.6). Fig. 3.15 depicts trajectory of the vortex center
for different x01 in (2.6) when ε = 1
32in and dε1 for different ε. From Fig. 3.15 and
3.3 Numerical results under Neumann BC 48
(a) -1 0 1-1
0
1
x
y
0 0.46 0.920
0.14
0.28
t
d 1ε
ε=1/25
ε=1/50
ε=1/100
(b) -1 0 1-1
0
1
x
y
0 0.56 1.120
0.2
0.4
t
d 1ε
ε=1/25
ε=1/50
ε=1/100
Figure 3.15: Trajectory of the vortex center when ε = 132
(left column) and dε1 for dif-
ferent ε (right column) for the motion of a single vortex in GLE under homogeneous
Neumann BC with different x01 in (2.6): (a) x0
1 = (0, 0.1), (b) x01 = (0.1, 0.1).
additional numerical results not shown here for brevity, we can see that: (i). The
initial location of the vortex, i.e., value of x0 affects the motion of the vortex a lot
and this shows the effect on the vortex from the Neumann BC. (ii). If x01 = (x0, y0) 6=
(0, 0) satisfies x0 = 0 or y0 = 0 or x0 = ±y0, the trajectory is a straight line. (iii).
If x01 = (0, 0), the vortex will not move all the time, otherwise, the vortex will move
and finally exit the domain and never come back. This is quite different from the
situations in bounded domain with Dirichlet BC where a single vortex can never
move outside the domain or in the whole space where a single vortex doesn’t move
at all under the initial condition (2.6) when D = R2. (iv). As ε → 0, the dynamics of
the vortex center under the GLE dynamics converges uniformly in time to that of the
reduced dynamical laws well before it exits the domain, which verifies numerically
the validation of the reduced dynamical laws in this case. Of course, when the vortex
center is being exited the domain or after it moves out of the domain, the reduced
dynamics laws are no longer valid. However, the dynamics of GLE is still physically
interesting. In fact, based on our extensive numerical experiments, the motion of
the vortex centers from the reduced dynamical law agrees with that from the GLE
dynamics qualitatively when 0 < ε < 1 and quantitatively when 0 < ε ≪ 1 well
before it moves out of the domain.
3.3 Numerical results under Neumann BC 49
3.3.2 Vortex pair
Here we present numerical results of the interaction of vortex pair under the
GLE dynamics and its corresponding reduced dynamical laws, i.e., we take M = 2,
n1 = n2 = 1, x01 = (−0.5, 0) and x0
2 = (0.5, 0) in (2.6).
Fig. 3.16 depicts time evolution of the amplitude |ψε|, time evolution of the
GL functionals, xr1(t), xε1(t) and d
ε1(t), and trajectory of the vortex centers for GLE
under homogeneous Neumann BC.
From Fig. 3.16 and additional numerical results not shown here for brevity,
we can draw the following conclusions for the interaction of vortex pair under the
GLE dynamics with homogeneous Neumann BC: (i). The two vortices undergo a
repulsive interaction, their centers move outwards along the line connected them
initially with symmetric trajectories, i.e., xε1(t) = −xε2(t) (cf. Fig. 3.16 (a) & (b)).
Moreover, if the two vortices are not located symmetrically initially, the one closer
to the boundary will first move outside the domain and the other one will exit
the domain later. All the vortices will exit the domain D at finite time Tε which
increases when ε decreases. (ii). When ε → 0, the dynamics of the two vortex
centers under the GLE dynamics converge uniformly in time to that of the reduced
dynamical laws before any one of them exit the domain (cf. Fig. 3.16(c)), which
verifies numerically the validation of the reduced dynamical laws in this case. In
fact, based on our extensive numerical experiments, the motion of the two vortex
centers from the reduced dynamical laws agree with those from the GLE dynamics
qualitatively when 0 < ε < 1 and quantitatively when 0 < ε ≪ 1. (iii). During
the dynamics of GLE, the GL functional and its kinetic parts decrease when time
increases, its interaction part doesn’t change much when t is small and changes
dramatically when any one of the two vortices move outside the domain D. When
t→ ∞, all the three quantities converge to 0 (cf. Fig. 3.16(c)), which imply that a
constant steady state will be reached in the form of φε(x) = eic0 for x ∈ D with c0
a constant.
3.3 Numerical results under Neumann BC 50
(a)
(b)-1 0 1
-1
0
1
x
y
0 0.1 0.20
13
26
t
Total Energy
Kinetic Energy
0 0.1 0.20
1.7
3.4
t
Interaction Energy
(c) 0 0.06 0.12-1
-0.75
-0.5
t
x 1 ε(t)
or x
1 r(t)
ε=1/32
ε=1/100
reduced
0 0.06 0.12-0.1
0
0.1
t
y 1 ε(t)
or y
1 r (t)
ε=1/32
ε=1/100
reduced
0 0.05 0.10
0.08
0.16d 1ε
t
ε=1/32
ε=1/64
ε=1/100
Figure 3.16: Dynamics and interaction of a vortex pair in GLE under Neumann
BC: (a) contour plots of |ψε(x, t)| with ε = 132
at different times, (b) trajectory of the
vortex centers (left) and time evolution of the GL functionals (right) for ε = 132, (c)
time evolution of xε1(t) and xr1(t) (left and middle) and their difference dε1(t) (right)
for different ε.
3.3.3 Vortex dipole
Here we present numerical results of the interaction of vortex dipole under the
GLE dynamics and its corresponding reduced dynamical laws, i.e., we take M = 2,
n1 = −1, n2 = 1, x01 = (−d0, 0) and x0
2 = (d0, 0) with d0 a constant.
Fig. 3.17 depicts contour plots of the amplitude |ψε|, while Fig. 3.18 shows time
evolution of GL functionals and trajectory of the vortex centers when ε = 132
in for
3.3 Numerical results under Neumann BC 51
(a)
(b)
Figure 3.17: Contour plots of |ψε(x, t)| at different times for the interaction of vortex
dipole in GLE under Neumann BC with ε = 132
for different d0: (a) d0 = 0.2, (b)
d0 = 0.7.
(a) -0.2 0 0.2
-0.2
0
0.2
x
y 0 0.07 0.140
9
18
t
0 0.07 0.140
1.6
3.2
t
E ε
int
E ε
E ε
kin
(b) -1 0 1-1
0
1
x
y
0 0.11 0.220
10
20
t
0 0.11 0.220
1.7
3.4
t
E ε
E ε
kin
E ε
int
Figure 3.18: Trajectory of vortex centers (left) and time evolution of the GL func-
tionals (right) for the interaction of vortex dipole in GLE under Neumann BC with
ε = 132
for different d0: (a) d0 = 0.2, (b) d0 = 0.7.
different d0 in (2.6). Fig. 3.19 shows time evolution of xr1(t), xε1(t) and dε1(t) for
different ε and d0.
From Figs. 3.17, 3.18 & 3.19 and additional numerical results not shown here for
brevity, we can draw the following conclusions for the interaction of vortex dipole
3.3 Numerical results under Neumann BC 52
0 0.011 0.022-0.2
-0.1
0
t
x 1 ε(t
) or
x 1 r
(t)
ε=1/32
ε=1/128
reduced
0 0.011 0.022-0.1
0
0.1
t
y 1 ε(t
) or
y1 r (t
)
ε=1/32
ε=1/128
reduced
0 0.008 0.0160
0.07
0.14
d 1ε
t
ε=1/32
ε=1/64
ε=1/128
0 0.027 0.055-0.1
0
0.1
t
y 1 ε(t
) or
y1 r (t
)
ε=1/32
ε=1/128
reduced
0 0.027 0.055-1
-0.85
-0.7
t
x 1 ε(t
) or
x 1 r
(t)
ε=1/32
ε=1/128
reduced
0 0.027 0.0550
0.07
0.14
dε 1
t
ε=1/32
ε=1/64
ε=1/128
Figure 3.19: Time evolution of xε1(t) and xr1(t) (left and middle) and their difference
dε1(t) (right) for different ε and d0: (a) d0 = 0.2, (b) d0 = 0.7.
under the GLE dynamics with homogeneous Neumann BC: (i). The initial location
of the vortices, i.e., d0, affects the motion of vortices significantly. In fact, there exists
a critical value drc = dεc for 0 < ε < 1, which is found numerically as drc = 0.5, such
that when the distance between the two vortex centers initially d0 =12|x0
1−x02| < drc,
then the two vortices will move towards each other along the line connecting their
initial locations and finally merge at the origin at finite time Tε which increases when
ε decreases, and respectively, when d0 > drc, the two vortices will move outwards
along the line connecting their initial locations and finally move out of the domain at
finite time Tε which increases when ε decreases (cf. Figs. 3.17 & 3.18). Moreover, the
trajectories of the two vortices are symmetric, i.e., x1(t) = −x2(t), and finally the
GLE dynamics will lead to a constant steady state with amplitude 1, i.e., φε(x) = eic0
for x ∈ D with c0 a real constant. (ii). When ε → 0, the dynamics of the two vortex
centers under the GLE dynamics converges uniformly in time to that of the reduced
dynamical laws before they collide or move out of the domain (cf. Fig. 3.19) which
verifies numerically the validation of the reduced dynamical laws in this case. In
3.3 Numerical results under Neumann BC 53
fact, based on our extensive numerical experiments, the motion of the two vortex
centers from the reduced dynamical laws agree with those from the GLE dynamics
qualitatively when 0 < ε < 1 and quantitatively when 0 < ε ≪ 1. (iii). During
the dynamics of GLE, the GL functional and its kinetic part decrease when time
increases, its interaction part doesn’t change much when t is small. All the three
quantities changes dramatically when the two vortices collide or move across ∂Dand eventually converge to 0 when t→ ∞ (cf. Fig. 3.18).
3.3.4 Vortex cluster
Here we present numerical results of the interaction of vortex clusters under
the GLE dynamics. We will consider the following cases: case I. M = 3, n1 =
n2 = n3 = 1, x01 = (−0.2,
√35), x0
2 = (−0.2,−√35), x0
3 = (0.4, 0); case II. M = 3,
n1 = n2 = n3 = 1, x01 = (−0.4, 0), x0
2 = (0, 0), x03 = (0.4, 0); case III. M = 3,
n1 = n2 = n3 = 1, x01 = (−0.4, 0.2), x0
2 = (0, 0.2), x03 = (0.4, 0.2); case IV. M = 3,
n2 = −1, n1 = n3 = 1, x01 = (−0.4, 0), x0
2 = (0, 0), x03 = (0.4, 0); case V. M = 3,
n3 = −1, n1 = n2 = 1, x01 = (−0.2,
√35), x0
2 = (−0.2,−√35), x0
3 = (0.4, 0); case VI.
M = 4, n1 = n2 = n3 = n4 = 1, x01 = (0.4,−0.4 sin(1)), x0
2 = (−0.2, 0.4 cos(1)),
x03 = (−0.2, 0.4 sin(1)), x0
4 = (0, 0); case VII. M = 4, n1 = n3 = −1, n2 = n4 = 1,
x01 = (−0.4, 0), x0
2 = (− 215, 0), x0
3 = ( 215, 0), x0
4 = (0.4, 0); case VIII. M = 4,
n1 = n2 = −1, n3 = n4 = 1, x01 = (0.4, 0), x0
2 = (−0.2,√35), x0
3 = (−0.2,−√35),
x04 = (0, 0); and case IX. M = 4, n2 = n3 = 1, n1 = n4 = −1, x0
1 = (0.4, 0),
x02 = (−0.2,
√35), x0
3 = (−0.2,−√35), x0
4 = (0, 0).
Fig. 3.20 shows trajectory of the vortex centers when ε = 132
in for the above 9
cases. From Fig. 3.20 and additional numerical results not shown here for brevity, we
can draw the following conclusions: (i). The interaction of vortex clusters under the
GLE dynamics with homogeneous Neumann BC is very interesting and complicated.
The detailed dynamics and interaction pattern of a vortex cluster depends on its
initial alignment of the vortices and geometry of the domain D. (ii). For a cluster
of M vortices, if they have the same index, then at least M − 1 vortices will move
3.3 Numerical results under Neumann BC 54
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 -0.3 0.4-0.7
0
0.7
x
y
-0.4 0 0.4-0.4
0
0.4
x
y
-0.4 0 0.4
-0.4
0
0.4
x
y
-0.35 0 0.4-0.4
0
0.4
x
y
-0.34 0 0.45-0.4
0
0.4
x
y
Figure 3.20: Trajectory of vortex centers for the interaction of different vortex
clusters in GLE under homogeneous Neumann BC with ε = 132
for cases I-IX (from
left to right and then from top to bottom) in section 3.3.4.
out of the domain at finite time and no collision will happen for any time t ≥ 0.
On the other hand, if they have opposite index, collision will happen at finite time.
After collisions, the leftover vortices will then move out of the domain at finite time
and at most one vortex may left in the domain. When t is sufficiently large, in most
cases, no vortex is left in the domain; of course, when the geometry and initial setup
are properly symmetric and M is odd, there maybe one vortex left in the domain.
3.3 Numerical results under Neumann BC 55
Figure 3.21: Contour plots of the amplitude |ψε(x, t)| for the initial data (top)
and corresponding steady states (bottom) of vortex cluster in the GLE under ho-
mogeneous Neumann BC with ε = 116
for different number of vortices M and
winding number nj : M = 3, n1 = n2 = n3 = 1 (first and second columns);
M = 3, n1 = −n2 = n3 = 1 (third column); and M = 4, n1 = −n2 = n3 = −n4 = 1
(fourth column).
3.3.5 Steady state patterns of vortex clusters
Here we present the steady state patterns of vortex clusters under the GLE
dynamics with homogeneous Neumann BC. To this end, we take ε = 116
in and
assume the M vortices are initially located uniformly on a line, i.e.,
x0j =
(−0.5 +
j − 1
M − 1, 0
), j = 1, 2, . . . ,M,
or on a circle with radius R1 = 0.5, i.e.,
x0j = 0.5
(cos
(2jπ
M
), sin
(2jπ
M
)), j = 1, 2, . . . ,M.
Fig. 3.21 depicts the amplitude |ψε| of the initial data and final steady states under
the GLE dynamics with different initial setups.
3.3 Numerical results under Neumann BC 56
From Fig. 3.21 and additional numerical results not shown here for brevity,
we can draw the following conclusions for the interaction of vortex clusters under
the GLE dynamics with homogeneous Neumann BC: (i). If the three like vortices
initially located uniformly on a circle, they will repel each other and finally exit
outside the domain and never come back. (ii). If the three like vortices initially
located uniformly on a line, the left and right vortices will finally exit outside the
domain and never come back, while the middle one does not move all the time. (iii).
If the three vortices initially located uniformly on a line with the middle vortex
whose winding number is opposite to the other two, the middle one will not move
all the time, while the left and right vortices will move toward the origin and one
of them will merge with the middle vortex, finally only one vortex will stay at the
origin forever. (iv). If the four vortices initially located uniformly on a circle with
the sign of winding number alternatively changed, the four vortices will move toward
the original point and merge with each other, and finally there will be no vortex
in the domain. (v). Actually, from our extensive numerical experiments, we can
conclude that for any initial setup, if the number of vortices M is even, the vortices
will either merge or move outside the domain, and finally there will be no vortex
leftover in the domain; while if M is odd, there will be at most one vortex leftover
in the domain when t→ ∞.
3.3.6 Validity of RDL under small perturbation
Same as the motivation in section 3.2.6, here we study the radiation property
of the GLE dynamics under homogeneous Neumann BC in this subsection. To this
end, we take the initial data (1.2) as (3.4) with ψε0 chosen as (2.6) with M = 2,
n1 = n2 = 1, x01 = −x0
2 = (0.5, 0) and h(x) as (3.3). Then we take δ = ε and let ε go
to 0, and solve the GLE with initial condition (3.4) for the vortex centers xδ,ε1 (t) and
xδ,ε2 (t) and compare them with those from the reduced dynamical law. We denote
dδ,εj (t) = |xδ,εj (t)−xrj(t)| for j = 1, 2 as the error. Fig. 3.22 depicts time evolution of
dδ,ε1 (t) for the case when δ = ε, i.e., small perturbation, and the case when δ = 0, i.e.,
3.4 Vortex dynamics in inhomogeneous potential 57
0 0.05 0.10
0.085
0.17
t
d1δ,
ε
0 0.05 0.10
0.08
0.16
d1δ,
ε
t
δ=ε=1/32
δ=ε=1/64
δ=ε=1/100
δ=0,ε=1/32
δ=0,ε=1/64
δ=0,ε=1/100
Figure 3.22: Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 3.3.6
no perturbation. From this figure, we can see that small perturbation in the initial
data does not affect the motion of the vortices much, same as the non-perturbed
initial setups, the dynamics of the two vortex centers under the GLE dynamics with
perturbed initial setups also converge to those obtained from the reduced dynamical
law when ε→ 0, which is simply similar as the situation in the GLE dynamics with
perturbed initial data under Dirichlet BC.
3.4 Vortex dynamics in inhomogeneous potential
In this subsection, we study numerically the vortex dynamics in the GLE dynam-
ics under Dirichlet BC in inhomogeneous potential. We let the external potential in
(1.1) as:
V (x) =1
1 + 9e−γx((x−x0c)2−γy(y−y0c )2), x ∈ D, (3.5)
where γx, γy, x0c and y
0c are constants, i.e., we impose a single well external potential
with minimal location sitting at point (x0c , y0c ). To study the joint effect of the
pinning effect, the boundary effect and the interaction between vortices on the vortex
dynamics, we consider two types of inhomogeneous external potential:
• Type I. Symmetric external potential, i.e., γx = γy = 20;
• Type II. Anisotropic external potential, i.e., γx = 30, γy = 15.
3.4 Vortex dynamics in inhomogeneous potential 58
Choose the initial data as (2.6) and for simplicity, we always let h(x) = 0 and
(x0c , y0c ) = (0, 0) =: O. We study following three cases: case I: M = 1, n1 = 1, x1 =
[0.4, 0.4], V (x) is chosen as type I potential; case II: M = 1, n1 = 1, x1 = [0.4, 0.4],
V (x) is chosen as type II potential; case III: M = 2, n1 = n2 = 1, x1 = [−0.3, 0],
x2 = [0, 0.3], V (x) is chosen as type I potential.
Fig. 3.23, shows the trajectory, time evolution of the distance between the vortex
center and potential center and dε1(t) for different ε for case I and II, as well as
trajectory of vortex center for different ε of the vortices for case III. From this figure
and additional numerical experiment not shown here for brevity, we can see that:
(i). For the single vortex, it moves monotonically toward the points xp = (x0c , y0c ),
where the external potential V(x) attains its minimum value (cf. Fig. 3.23 (a)
& (b)), which shows clearly the pinning effect. Moreover, the trajectory depend
on the type of the potential V (x). The speed that vortex move to xp as well as
the final location that vortex stay steady depend on the value of ε (cf. Fig. 3.23
(a) & (b)). The smaller the ε is, the closer the final location to xp and the faster
the vortex move to it. (ii). As ε → 0, the dynamics of the vortex center in the
GLE dynamics converges uniformly in time to that in the reduced dynamics which
verifies numerically the validation of the reduced dynamical laws in this case. (iii).
For the vortex pair, when ε is large, the two vortices will move apart from each other
for a while, then monotonically toward each other and to xp, which illustrate the
pinning effect clearly (cf. Fig. 3.23 (c)). Then, they will move apart from each other
again in the opposite direction with one toward xp due to the repulsive interaction
between the two vortices. Otherwise, when ε is small enough, the vortex will simply
move monotonically close to each other and to xp and finally they will stop and
stay steady at someplace near xp. The smaller the ε is, the closer the two vortex
to the point xp. As we know, in the speeded time scale λε =1
ln( 1ε), the vortex pair
undergo a repulsive interaction and they always move apart from each other toward
the boundary in the GLE dynamics under homogeneous potential when h(x) = 0,
and the smaller the ε is, the stronger they repulse from each other, thus, it might
3.5 Conclusion 59
(a)0 0.2 0.4
0
0.2
0.4
x
y
ε=1/16
ε=1/64reduced
0 0.25 0.50
0.3
0.6
t
|xε 1−
O|
ε=1/16
ε=1/25
ε=1/64
0 0.36 0.720
0.16
0.32
t
d 1ε
ε=1/25ε=1/40ε=1/64
(b)0 0.2 0.4
0
0.2
0.4
x
y
ε=1/16
ε=1/64reduced
0 0.41 0.820
0.3
0.6
t
|xε 1−
O|
ε=1/16
ε=1/32
ε=1/64
0 0.41 0.820
0.22
0.44
t
d 1ε
ε=1/25
ε=1/40
ε=1/64
(c) -0.4 -0.05 0.3-0.3
0.05
0.4
x
y
ε=1/8
-0.4 -0.05 0.3-0.3
0.05
0.4
x
y
ε=1/16
-0.4 -0.05 0.3-0.3
0.05
0.4
x
yε=1/25
Figure 3.23: (a) and (b): trajectory, time evolution of the distance between the
vortex center and potential center and dε1(t) for different ε for case I and II, and (c):
Trajectory of vortex center for different ε of the vortices for case III in section 3.4.
be interesting that we studying the pinning effect of the vortex pair in speeded time
scale to see how the vortex pair continue to move after they move close to xp. Will
they move apart from each other again?
3.5 Conclusion
In this chapter, by applying the efficient and accurate numerical methods pro-
posed in chapter 2 to simulate Ginzburg-Landau equation (GLE) with a dimension-
less parameter 0 < ε < 1 on bounded domains with either Dirichlet or homogenous
3.5 Conclusion 60
Neumann BC and its corresponding reduced dynamical laws (RDLs), we studied
numerically quantized vortex interaction in GLE with/without impurities for super-
conductivity and compared numerically patterns of vortex interaction between the
GLE dynamics and its corresponding reduced dynamical laws under different initial
setups.
For the GLE under a homogeneous potential, based on extensive numerical re-
sults, we verified that the dynamics of vortex centers under the GLE dynamics
converges to that of the reduced dynamical laws when ε → 0 before they collide
and/or move out of the domain. Certainly, after either vortices collide with each
other or move out of the domain, the RDLs are no longer valid; however, the dy-
namics and interaction of quantized vortices are still physically interesting and they
can be obtained from the direct numerical simulations for the GLE with fixed ε > 0
even after they collide and/or move out of the domain. We also identified the pa-
rameter regimes where the RDLs agree with qualitatively and/or qualitatively as
well as fail to agree with those from the GLE dynamics. In the validity regimes, the
RDL is still valid under small perturbation in the initial data due to the dissipative
nature of the GLE. Some very interesting nonlinear phenomena related to the quan-
tized vortex interactions in the GLE for superconductivity were also observed from
our direct numerical simulation results of GLE. Different steady state patterns of
vortex clusters under the GLE dynamics were obtained numerically. From our nu-
merical results, we observed that boundary conditions and domain geometry affect
significantly on vortex dynamics and interaction, which showed different interaction
patterns compared to those in the whole space case [160, 161].
For the GLE in an inhomogeneous potential under the Dirichlet BC, we also
numerically verified the validity of the RDL. By directly simulating the GLE, we
find that vortices move in quite different ways from that in the homogeneous case.
The vortices basically move toward critical points of the inhomogeneous potential
in the limiting process ε → 0, which show the pinning effect that caused by the
impurities given by the inhomogeneities.
Chapter 4
Vortex dynamics in NLSE
In this chapter, we apply the numerical method presented in chapter 2 to simulate
quantized vortex interaction of NLSE, i.e., λε = 0, β = 1 in the GLSE (1.1), with
different ε and under different initial setups including single vortex, vortex pair,
vortex dipole and vortex cluster. We study how the dimensionless parameter ε,
initial setup, boundary value and geometry of the domain D affect the dynamics
and interaction of vortices. Moreover, we compare the results obtained from the
NLSE with those from the corresponding reduced dynamical laws, and identify the
cases where the reduced dynamical laws agree qualitatively and/or quantitatively
as well as fail to agree with those from NLSE on vortex interaction. Finally, we
also investigate the sound-vortex interaction and study the radiative nature of the
vortices in NLSE dynamics.
Without specification, the initial data is chose as the same one in section 3.1 in
chapter 3.
4.1 Numerical results under Dirichlet BC
4.1.1 Single vortex
In this subsection, we present numerical results of the motion of a single quan-
tized vortex in the NLSE dynamics and the corresponding reduced dynamics, i.e
61
4.1 Numerical results under Dirichlet BC 62
we take M = 1, n1 = 1 in (2.6). To study how the initial phase shift h(x) and
initial location of the vortex x0 affect the motion of the vortex and to understand
the validity of the reduced dynamical law, we consider the following 11 cases:
• Case I-III: x01 = (0, 0), and h(x) is chosen as Mode 1, 2 and 3, respectively;
• Case IV-VIII: x01 = (0.1, 0), and h(x) is chosen as Mode 1, 2, 3, 4 and 5,
respectively;
• Case IX-XI: x01 = (0.1, 0.1), and h(x) is chosen as Mode 3, 4 and 5, respectively.
Moreover, to study the effect of domain geometry, we consider D of three types: type
I. a square D = [−1, 1] × [−1, 1], type II. a rectangle D = [−1, 1] × [−0.65, 0.65],
and type III. a unit disk D = B1(0). Thus we also study the following 4 additional
cases:
• Case XII-XIII: x01 = (0, 0), h(x) = x + y, D is chosen as type II and III,
respectively;
• Case XIV-XV: x01 = (0.1, 0), h(x) = x2 − y2, D is chosen as type II and III,
respectively.
Fig. 4.1 depicts trajectory of the vortex center when ε = 140
for Cases I-VI and dε1
with different ε for Cases I, V and VI. Fig. 4.2 shows trajectory of the vortex center
when ε = 164
in NLSE for cases VI-XI, while Fig. 4.3 shows that for Cases XII-XVII
when ε = 132. From Figs. 4.1-4.3 and additional numerical experiments not shown
here for brevity, we can draw the following conclusions: (i). When h(x) ≡ 0, the
vortex center doesn’t move and this is similar to the case in the whole space. (ii).
When h(x) = (x + by)(x − yb) with b 6= 0, the vortex does not move if x0
1 = (0, 0),
while it does move if x01 6= (0, 0) (cf. Fig. 4.1 Cases III and VI for b = 1). (iii).
When h(x) 6= 0 and h(x) 6= (x+ by)(x− yb) with b 6= 0, in general, the vortex center
does move. For the NLSE dynamics, there exists a critical value εc depending on
h(x), x01 and D such that if ε < εc, the vortex will move periodically in a close
4.1 Numerical results under Dirichlet BC 63
loop (cf. Fig. 4.1), otherwise their trajectory will not be a close loop. This differs
from the situation in the reduced dynamics significantly, where the trajectory is
always periodic (cf. Fig. 4.2 red dash line). Thus the reduced dynamical laws fail
qualitatively when ε > εc. It should be an interesting problem to find how this
critical value depends on h(x), x01 and the geometry of D. (iv). In general, the
initial location, the geometry of the domain and the boundary value will all affect
the motion of the vortex (cf. Fig. 4.3). (v). When ε → 0, the dynamics of the vortex
center in the NLSE dynamics converges uniformly in time to that in the reduced
dynamics (cf. Fig. 4.1 bottom row) which verifies numerically the validation of the
reduced dynamical laws.
4.1.2 Vortex pair
Here we present numerical results of the interaction of vortex pair under the
NLSE dynamics and its corresponding reduced dynamical laws, i.e., we take M = 2,
n1 = n2 = 1. We always suppose that the two vortices are initially located symmetric
on the x-axis, i.e., we take x02 = −x0
1 = (d0, 0) with 0 < d0 < 1 in (2.6). Let ε = 140,
we consider the following three cases: case I. d0 = 0.1 and h(x) = 0, case II. d0 = 0.5
and h(x) = 0, case III. d0 = 0.5 and h(x) = x+ y. Fig. 4.4 depicts the trajectory of
the vortex pair, the time evolution of Eε(t), Eεkin(t), xε1(t), xε2(t) and dε1(t) for above3 cases.
From Figs. 4.4 and additional numerical results not shown here for brevity, we
can draw the following conclusions for the interaction of vortex pair under the NLSE
dynamics with Dirichlet BC: (i). The total energy is conserved during the dynamics
of the NLSE in all cases. (ii). The pattern of the motion of the vortex centers depend
on both the initial location of the two vortices and the initial phase shift h(x) in
(2.6). (iii). When h(x) ≡ 0, the vortices move periodically and their trajectories
are symmetric, i.e., xε1(t) = −xε2(t). Moreover, for both the reduced dynamical law
and NLSE dynamics, there is a critical value of d0, say drc and d
εc respectively, such
that if d0 < drc (or d0 < dεc in NLSE dynamics), the two vortices will rotate with
4.1 Numerical results under Dirichlet BC 64
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
0 0.5 10
0.17
0.34
t
d 1ε
ε=1/32ε=1/50ε=1/80
0 0.8 1.60
0.06
0.12
t
d 1ε
0 0.5 10
0.2
0.4
t
d 1ε
ε=1/32ε=1/50ε=1/80
ε=1/16ε=1/40ε=1/80
Figure 4.1: Trajectory of the vortex center in NLSE under Dirichlet BC when ε = 140
for Cases I-VI (from left to right and then from top to bottom in top two rows),
and dε1 for different ε for Cases I,V&VI (from left to right in bottom row) in section
4.1.1.
each other and move along a circle-like trajectory, otherwise, they will move along
a crescent-like trajectory (cf. Figs.4.4 (a) & (b)). We find numerically the critical
value drc ≈ 0.4923 and dεc for 0 < ε < 1 which are depicted in Fig. 4.5. From these
values, we can fit the following relation for drc and dεc:
dεc = drc + 2.11ε2.08, 0 < ε < 1.
(iv). When h(x) 6= 0, it affects the motion of the two vortex centers significantly (cf.
Figs.4.4 (c)). (v). For any fixed d0, the dynamics of the two vortex centers under
4.1 Numerical results under Dirichlet BC 65
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
yFigure 4.2: Trajectory of the vortex center in NLSE dynamics under Dirichlet BC
when ε = 164
(blue solid line) and from the reduced dynamical laws (red dash line)
for Cases VI-XI (from left to right and then from top to bottom) in section 4.1.1.
the NLSE dynamics converges uniformly in time to that of the reduced dynamical
laws (cf. Figs.4.4 (d)) when ε→ 0. However, for fixed ε, the reduced dynamical law
fails qualitatively to describe the motion of vortices if drc < d0 < dεc.
4.1.3 Vortex dipole
Here we present numerical results of the interaction of vortex dipole under the
NLSE dynamics and its corresponding reduced dynamical laws, i.e., we take M = 2,
n1 = −n2 = −1, x02 = −x0
1 = (d0, 0) in (2.6) with d0 = 0.5 and ε = 125. Fig.
4.6 depicts contour plots of |ψε(x, t)| at different times, trajectory of the vortex
dipole, time evolution of xε1(t), xε2(t) and d
ε1(t) for different h(x). From Fig. 4.6 and
additional numerical results not shown here for brevity, we can draw the following
conclusions: (i). The total energy is conserved numerically very well during the
dynamics. (ii). The pattern of the motion of the vortex centers depend on both the
initial location of the two vortices and the initial phase shift h(x) in (2.6). (iii). The
4.1 Numerical results under Dirichlet BC 66
-1 0 1-0.65
0
0.65
x
y-1 0 1
-1
0
1
x
y
-1 0 1-0.65
0
0.65
x
y
-1 0 1-1
0
1
x
y
Figure 4.3: Trajectory of the vortex center in NLSE under Dirichlet BC when ε = 140
for cases I, XII-XIII, VI and XIV-XV (from left to right and then from top to bottom)
in section 4.1.1.
vortex dipole moves upward symmetrically with respect to y-axis and finally merges
and annihilates somewhere near the top boundary simultaneously. The distance
between the merging place and the boundary is of O(ε) when ε is small. After
merging, new waves will be created and reflected by the top boundary. The new
waves will then move back into the domain and be reflected back into the domain
again when they hit the boundaries (cf. Fig. 4.6). Moreover, the vortex dipole in
the NLSE dynamics will always merge in some place near the top boundary for all
d0. However, in the reduced dynamics, they never merge inside D, in fact, they will
move outside the domain before they merge. Hence, the reduced dynamical law fails
quantitatively when the vortex dipole is near the boundary. (iv). When h(x) 6= 0,
it affects the motion of the two vortex centers significantly (cf. Figs.4.4 (c)). (v).
When ε → 0, the dynamics of the two vortex centers under the NLSE dynamics
converges uniformly to that of the reduced dynamical laws (cf. Fig. 4.6) before they
merge each other or near the boundary which verifies numerically the validation of
4.1 Numerical results under Dirichlet BC 67
(a)-0.8 0 0.8
-0.8
0
0.8
x
y
0 0.075 0.1531.6
34.4
37.2
t
E εk i n
E ε
0 0.075 0.15-0.11
0.11
x 1ε or x
2ε
t
0 0.075 0.15-0.11
0.11
t
y 1ε or y
2ε
(b)-1 0 1
-1
0
1
x
y
0 2 423
25
27
x
y
Eεkin
Eε0 2 4
-1
1
t
x 1ε or x
2ε
0 2 4-1
1
t
y 1ε or y
2ε
(c)-1 0 1
-1
0
1
x
y
0 0.45 0.926.2
28.7
31.2
x
y
Eεkin
Eε
0 0.45 0.9-0.8
1
t
x 1ε or x
2ε
0 0.45 0.9-1
1
t
y 1ε or y
2ε
(d)0 0.075 0.15
0
0.1
0.2
t
d 1ε
ε=1/40ε=1/50ε=1/64
0 2 40
0.47
0.94
t
d 1ε
ε=1/16ε=1/25ε=1/40
0 0.45 0.90
0.43
0.86
t
d 1ε
ε=1/32ε=1/40ε=1/50
Figure 4.4: Form left to right in (a)-(c): trajectory of the vortex pair, time evolution
of Eε(t) and Eεkin(t) as well as xε1(t) and xε2(t) for the 3 cases in section 4.1.2. (a).
case I, (b). case II, (c). case III. (d). time evolution of dε1(t) for case I-III (form left
to right).
the reduced dynamical laws in this case. In fact, based on our extensive numerical
experiments, the motions of the two vortex centers from the reduced dynamical
4.1 Numerical results under Dirichlet BC 68
0.02 0.06 0.10
0.01
0.02
ε
d cε -dcr
-3.8 -2.8 -1.8-7
-5.5
-3.8
ln(ε)
ln(d
cε -dcr )
Figure 4.5: Critical value dεc for the interaction of vortex pair of the NLSE under
the Dirichlet BC with different ε and h(x) = 0 in (2.6): if d0 < dεc, the two vortex
will move along a circle-like trajectory, if d > dεc, the two vortex will move along a
crescent-like trajectory.
laws agree with those from the NLSE dynamics qualitatively when 0 < ε < 1 and
quantitatively when 0 < ε≪ 1 when they are not too close to the boundary.
4.1.4 Vortex cluster
Here we present numerical studies on the dynamics of vortex clusters in the
NLSE with Dirichlet BC (1.3), i.e., choose the initial data (1.2) as (2.6) and study
four cases:
Case I. M = 3, n1 = n2 = n3 = 1, x01 = −x0
3 = (d0, 0) and x02 = (0, 0).
Case II. M = 3, n1 = −n2 = n3 = 1, x01 = −x0
3 = (d0, 0) and x02 = (0, 0).
Case III. M = 4, n1 = n2 = −n3 = −n4 = 1, x01 = −x0
2 = (d1, 0) and x03 =
−x04 = (0, d2) with 0 < d1, d2 < 1.
Case IV. D = B5(0),M = 9, n1 = n2 = · · · = n9 = 1 and the 9 vortex centers are
initially located on a 3×3 uniform mesh points for the rectangle [−d0, d0]× [−d0, d0]with 0 < d0 < 1.
Fig. 4.7 depicts the trajectory and time evolution of x01(t), x
02(t) and x0
3(t) in the
NLSE dynamics for Cases I and II. Fig. 4.8 shows contour plots of |ψε| at differenttimes in the NLSE dynamics for Case III, and Fig. 4.9 depicts contour plots of−|ψε|,Sε(x, t) as well as slice plots of |ψε(x, 0, t)| for showing sound wave propagation of
4.1 Numerical results under Dirichlet BC 69
-1 0 1-1
0
1
x
y
0 0.15 0.30
0.04
0.08
t
d 1ε
ε=1/8
ε=1/16
ε=1/32
0 0.165 0.33-0.6
0.6
t
x 1ε or
x 2ε
0 0.165 0.330
1
t
y 1ε or
y 2ε
-1 0 1-1
0
1
x
y
0 1.25 2.5-1
1
t
x 1ε or
x 2ε
0 1.25 2.5-0.7
0.7
t
y 1ε or
y 2ε
0 1.25 2.50
0.2
0.4
t
d 1ε
ε=1/8ε=1/16ε=1/25
Figure 4.6: Contour plots of |ψε(x, t)| at different times (top two rows) as well as
the trajectory, time evolution of xε1(t), xε2(t) and dε1(t) (bottom two rows) for the
dynamics of a vortex dipole with different h(x)in section 4.1.3: (1). h(x) = 0 (top
three rows), (2). h(x) = x+ y (bottom row).
the NLSE dynamics in Case IV. Based on Figs. 4.7-4.9 and additional computations
not shown here for brevity, we can draw the following conclusions: (1). For Case I,
the middle vortex (initially at the origin) will not move while the other two vortices
4.1 Numerical results under Dirichlet BC 70
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-0.4 0 0.4-0.4
0
0.4
x
y
0 0.15 0.31 0.456-0.3
0
0.3
0.6
0.9
t
x
1ε(t)
y1ε(t)
x2ε(t)
y2ε(t)
0.456 0.49 0.53 0.56-0.3
0
0.3
0.6
0.9
t
x
1ε(t)
y1ε(t)
x2ε(t)
y2ε(t)
0.56 0.89 1.21 1.54-0.3
0
0.3
0.6
0.9
t
x
1ε(t)
y1ε(t)
x2ε(t)
y2ε(t)
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
0 0.37 0.73 1.1-0.3
0
0.3
0.6
0.9
t
x3ε(t)
y3ε(t)
x2ε(t)
y2ε(t)
1.1 1.15 1.2 1.25-0.5
0
0.5
1
t
x
3ε(t)
y3ε(t)
x2ε(t)
y2ε(t)
1.25 1.83 2.42 3-0.2
0
0.2
0.4
t
x
1ε(t)
y1ε(t)
Figure 4.7: Trajectory of the vortex xε1 (blue line), xε2 (dark dash-dot line) and xε3
(red dash line) (first and third rows) and their corresponding time evolution (second
and fourth rows) for Case I (top two rows) and Case II (bottom two rows) for small
time (left column), intermediate time (middle column) and large time (right column)
with ε = 140
and d0 = 0.25 in section 4.1.4.
4.1 Numerical results under Dirichlet BC 71
Figure 4.8: Contour plots of |ψε(x, t)| with ε = 116
at different times for the NLSE
dynamics of a vortex cluster in Case III with different initial locations: d1 = d2 =
0.25 (top two rows); d1 = 0.55, d2 = 0.25 (middle two rows); d1 = 0.25, d2 = 0.55
(bottom two rows) in section 4.1.4.
4.1 Numerical results under Dirichlet BC 72
(a) (b)
(c) (d)
(e)
-5 0 50
0.56
1.12 t=0
x
|ψε (x
,0,t)
|
-5 0 50
0.56
1.12 t=0.057
x
|ψε (x
,0,t)
|
-5 0 50
0.56
1.12 t=0.174
x
|ψε (x
,0,t)
|
-5 0 50
0.56
1.12 t=0.291
x
|ψε (x
,0,t)
|
(f)
-5 0 50
0.54
1.08 t=0.372
x
|ψε (x
,0,t)
|
-5 0 50
0.54
1.08 t=0.588
x
|ψε (x
,0,t)
|
-5 0 50
0.54
1.08 t=0.777
x
!ψε (x
,0,t)
|
-5 0 50
0.54
1.08 t=0.969
x
|ψε (x
,0,t)
|
Figure 4.9: Contour plots of −|ψε(x, t)| ((a) & (c)) and the corresponding phase
Sε(x, t) ((b) & (d)) as well as slice plots of |ψε(x, 0, t)| ((e) & (f)) at different times
for showing sound wave propagation under the NLSE dynamics of a vortex cluster
in Case IV with d0 = 0.5 and ε = 18in section 4.1.4.
rotate clockwise around the origin for some time. This dynamics agrees very well
with the NLSE dynamics in the whole plane [160, 161]. After some time, the above
4.1 Numerical results under Dirichlet BC 73
0 0.05 0.10
0.03
0.06
t
d 1δ,ε
δ=0, ε=1/64
δ=0, ε=1/80
δ=0, ε=1/100
0 0.05 0.10
0.008
0.016
t
d 1δ,ε
δ=ε=1/64
δ=ε=1/80
δ=ε=1/100
Figure 4.10: Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 4.1.5
symmetric structure is broken due to boundary effect and numerical errors, i.e.
the middle vortex will begin to move towards one of the other two vortices and
form as a vortex pair which will rotate with each other and also with the other
single vortex for a while. Then this pair will separate and one of them will form
a new vortex pair with the single vortex, leave the other one to be a new single
vortex rotating with them. This process will be repeated tautologically like three
dancers exchange their partners alternatively. In fact, the dynamical pattern after
the symmetric structure breaking depends highly on the mesh size and time step.
(ii). For Case II, similar as Case I, the middle vortex (initially at the origin) will
not move while the other two vortices rotate counter-clockwise around the origin for
some time. This dynamics agrees very well with the NLSE dynamics in the whole
plane [160, 161]. After some time, again the above symmetric structure is broken
due to boundary effect and numerical errors, i.e. the middle vortex will begin to
move towards one of the other two vortices and form as a vortex dipole which will
move nearly parallel towards the boundary and merge near the boundary. Sound
waves will be created and reflected back into the domain which drive the leftover
vortex in the domain to move (cf. Fig. ??). Again the dynamical pattern after
the symmetric structure breaking depends highly on the mesh size and time step.
From section 4.1.1, we know that a single vortex in the NLSE with h(x) = 0 does
not move, hence this example illustrates clearly the sound-vortex interactions which
cannot be predicted by the reduced dynamical laws since they are only valid before
annihilation and when ε is very small. (iii). For Case III, the four vortices form
4.1 Numerical results under Dirichlet BC 74
as two vortex dipoles when t is small. Then the two dipoles will move outwards
in opposite direction and finally the two vortices in each vortex dipole merge and
annihilate at some place near the boundary. If d1 = d2, the two vortex dipoles
move symmetrically with respect to the line y = x, and merge some place near the
top-right and bottom-left corners, respectively; if d1 > d2, both of them will move
towards the top-bottom boundary and merge near there; and if d1 < d2, both of
them will move towards the side boundary and merge near there. New waves are
created after merging and they are reflected back into the domain when they hit
the boundary (cf. Fig. 4.8). (iv). For Case IV, the vortex initially at the origin
does not move due to symmetry, while the other eight vortices rotate clockwise and
move along two circle-like trajectories (cf. Fig. 4.9). During the dynamics, sound
waves are generated and they propagate outwards and are reflected back into the
domain when they hit the boundary. The distances between other vortices and the
one centered at the origin increase when sound waves are radiated outwards; on the
other hand, they decrease and become even smaller than their initial distances when
sound waves are reflected by the boundary and move back into the domain (cf. Fig.
4.9). This example clearly shows sound waves and their impact on the dynamics of
vortices.
(v). Similar to the case in BEC setups [92,113,123,146], the symmetric structure
near t = 0 in Cases I & II for 3 vortices is dynamically unstable, i.e. symmetric
structure breaking will happen very quickly if we perturb the location of either the
central vortex or one of the side vortices a little bit. To show this, we perturb the
initial locations of some vortices in Case I to:
Type I. Asymmetric perturbation: x02 from (0, 0) → (δ, 0) with δ > 0;
Type II. Asymmetric perturbation: x01 from (d0, 0) → (d0 − δ, 0) with δ > 0;
Type III. Symmetric perturbation: x01 from (d0, 0) → (d0 − δ, 0) and x0
3 from
(−d0, 0) → (−d0 + δ, 0) with δ > 0.
Fig. 4.11 depicts the trajectory of xε1(t), xε2(t) and xε3(t) as well as d
ε2(t) := |xε2(t)|
for the above three perturbations. From this figure and additional computations
4.1 Numerical results under Dirichlet BC 75
not shown here, we see that the symmetric structure is dynamically stable under a
symmetric perturbation, and it is dynamically unstable under a small asymmetric
perturbation. For more stability analysis of vortex clusters, we refer to [27, 92, 113,
117, 123, 145, 146] and references therein.
-0.3 0 0.3-0.3
0
0.3
x
y
tra jectory for t ∈ [0,0.1]
-0.3 0 0.3-0.3
0
0.3
x
y
tra jectory for t ∈ [0,0.1]
-0.3 0 0.3-0.3
0
0.3
x
y
tra jectory for t ∈ [0,0.52]
0 0.06 0.120
0.14
0.28
t
|xε 2(t)|
δ=0.004
δ=0.002
δ=0.001
δ=0.0005
0 0.06 0.120
0.14
0.28
t
|xε 2(t)|
δ=0.004δ=0.002δ=0.001δ=0.0005
0 0.29 0.580
0.15
0.3
t
|xε 2(t)|
δ=0.05δ=0.004δ=0.002δ=0.0005
Figure 4.11: Trajectory of the vortex xε1(t) (blue dash-line), xε2(t) (red solid line)
and xε3(t) (dark dash-dot line) (top row) for δ = 0.0005 as well as time evolution
of the distance of xε2 to origin, i.e. |xε2(t)|, (bottom row) for Type I (first column),
Type II (second column) and Type III (third column) perturbation on the initial
location of the vortices for different δ in section 4.1.4, where ε = 140
and d0 = 0.25.
4.1.5 Radiation and sound wave
We study the radiation property of the NLSE dynamics under Dirichlet BC in
this subsection. To this end, we study two types of perturbation.
Type I: Perturbation on the initial data, i.e., we take the initial data (1.2) as
ψε(x, 0) = ψδ,ε0 (x) = ψε0(x) + δe−10((|x|−0.08)2+y2), x = (x, y) ∈ D, (4.1)
4.1 Numerical results under Dirichlet BC 76
where ψε0 is given in (2.6) with h(x) ≡ 0, M = 2, n1 = n2 = 1 and x01 = −x0
2 =
(0.1, 0). Then we take δ = ε and let ε go to 0, and solve the NLSE with initial
condition (4.1) for the vortex centers xδ,ε1 (t) and xδ,ε2 (t) and compare them with those
from the reduced dynamical law. We denote dδ,εj (t) = |xδ,εj (t)− xrj(t)| for j = 1, 2 as
the error. Fig. 4.10 depicts time evolution of dδ,ε1 (t) for the case when δ = ε, i.e.,
small perturbation, and the case when δ = 0, i.e., no perturbation. From this figure,
we can see that the dynamics of the two vortex centers under the NLSE dynamics
converge to those obtained from the reduced dynamical law when ε → 0 without
perturbation (cf. Fig. 4.10 left). On the contrary, the two vortex centers under the
NLSE dynamics do not converge to those obtained from the reduced dynamical law
when ε → 0 with small perturbation (cf. Fig. 4.10 right). This clearly demonstrates
radiation and sound wave effect on vortices in the NLSE dynamics with Dirichlet
BC.
Type II: Perturbation by an external potential, i.e., we replace V (x) ≡ 1 in
NLSE by V (x, t) = 1−W (x, t) with
W (x, t) =
− sin(2t)2, t ∈ [0, 0.5],
0, t > 0.5,x ∈ D. (4.2)
The initial data is chosen as (2.6) with M = 1, n1 = 1, x01 = (0, 0), D = B5(0) and
ε = 14. In fact, the perturbation is introduced when t ∈ [0, 0.5] and is removed after
t = 0.5. Fig. 4.12 illustrates surface plots of −|ψε| and contour plots of Sε(x, t)
as well as the slice plots of ψε(x, 0, t) at different times for showing sound wave
propagation. From Fig. 4.12, we can see that the perturbed vortex configuration
rotates and radiates sound waves. This agrees well with some former prediction in
the whole plane, for example, in Lange and Schroers [98] for the case M = 2. The
waves will be reflected back into the domain when they hit the boundary and then
be absorbed by the vortex core. Then the vortex core will radiate new waves and the
process is repeated tautologically. This process explicitly illustrates the radiation in
the NLSE dynamics.
Remark 4.1.1. Based on this example and other numerical results not show here for
4.1 Numerical results under Dirichlet BC 77
(a) (b)
(c) (d)
(e)
-5 0 50
0.54
1.08t=0
x
|ψε (x
,0,t)
|
-5 0 50
0.54
1.08t=0.22
x
|ψε (x
,0,t)
|
-5 0 50
0.54
1.08t=0.52
x
|ψε (x
,0,t)
|
-5 0 50
0.54
1.08t=0.6
x
|ψε (x
,0,t)
|
(f)
-5 0 50
0.54
1.08 t=1.1
x
|ψε (x
,0,t)
|
-5 0 50
0.59
1.18 t=1.56
x
|ψε (x
,0,t)
|
-5 0 50
0.59
1.18 t=2.4
x
|ψε (x
,0,t)
|
-5 0 50
0.63
1.26 t=2.97
x
|ψε (x
,0,t)
|
Figure 4.12: Surface plots of −|ψε(x, t)| ((a) & (c)) and contour plots of the corre-
sponding phase Sε(x, t) ((b) & (d)) as well as slice plots of |ψε(x, 0, t)| ((e) & (f)) at
different times for showing sound wave propagation under the NLSE dynamics in a
disk with ε = 14and a perturbation in the potential in section 4.1.5.
brevity, we can conclude that the vortex with winding number m = ±1 is dynamically
stable under the NLSE dynamics in a bounded domain with a perturbation in the
4.2 Numerical results under Neumann BC 78
-1 0 1-1
0
1
x
y
0 1.8 3.60
0.02
0.04
t
d 1ε
ε=1/16
ε=1/25
ε=1/32
-1 0 1-1
0
1
x
y
0 1.8 3.60
0.02
0.04
t
d 1ε
ε=1/8
ε=1/16
ε=1/25
Figure 4.13: Trajectory of the vortex center when ε = 132
and time evolution of dε1 for
different ε for the motion of a single vortex in NLSE under homogeneous Neumann
BC with x01 = (0.35, 0.4) (left two) or x0
1 = (0, 0.2) (right two) in (2.6) in section
4.2.1.
initial data and/or external potential. Meanwhile, we also found numerically that
the vortex with winding number m = 2 and ε = 132
is also dynamically stable under
a perturbation in the external potential. Actually, Mirionescu [115] indicated that
for a vortex with winding number |m| > 1, there exists a critical value εcm such
that if ε < εcm, the vortex is unstable, otherwise the vortex is stable. It was also
numerically observed that a vortex with |m| > 1 is unstable under a perturbation in
the potential but stable under a perturbation in the initial data in the whole plane
case [160]. Hence, it would be an interesting problem to investigate numerically
how the stability of a vortex depends on its winding number, value of ε and strength
and/or type of the perturbation under the NLSE dynamics in bounded domains.
4.2 Numerical results under Neumann BC
4.2.1 Single vortex
Here we present numerical results of the motion of a single quantized vortex
under the NLSE dynamics and its corresponding reduced dynamical laws, i.e., we
take M = 1 and n1 = 1 in (2.6). Fig. 4.13 depicts trajectory of the vortex center
for different x01 in (2.6) when ε = 1
32in NLSE and dε1 for different ε. From Fig. 4.13
and additional numerical results not shown here for brevity, we can see that:
(i). If x01 = (0, 0), the vortex will not move all the time, otherwise, the vortex will
4.2 Numerical results under Neumann BC 79
-1 0 1-1
0
1
x
y
0 0.92 1.8421
23
25
t
E εk i n
E ε
0 0.92 1.84-0.5
0.5
t
x 1ε or x
2ε
0 0.92 1.84-0.5
0.5
t
y 1ε or y
2ε
0 0.93 1.860
0.23
0.46
t
d 1ε
ε=1/8
ε=1/16
ε=1/32
Figure 4.14: Trajectory of the vortex pair (left), time evolution of Eε and Eεkin(second), xε1(t) and xε2(t) (third), and dε1(t) (right) in the NLSE dynamics under
homogeneous Neumann BC with ε = 132
and d0 = 0.5 in section 4.2.2.
move and its initial location x01 does not affect its motion qualitatively. Actually,
it moves periodically in a circle-like trajectory centered at the origin. This is quite
different from the situation in bounded domain with Dirichlet BC where the motion
of a single vortex depends significantly on its initial location for some h(x). It is also
quite different from the situation in the whole space where a single vortex doesn’t
move at all under the initial condition (2.6) when D = R2.
(ii). As ε → 0, the dynamics of the vortex center under the NLSE dynamics
converges uniformly in time to that of the reduced dynamical laws. In fact, based
on our extensive numerical experiments, the motion of the vortex center from the
reduced dynamical laws agrees with that from the NLSE dynamics qualitatively
when 0 < ε < 1 and quantitatively when 0 < ε ≪ 1.
4.2.2 Vortex pair
Here we present numerical results of the interaction of vortex pair under the
NLSE dynamics and its corresponding reduced dynamical laws, i.e., we take M = 2,
n1 = n2 = 1 and x02 = −x0
1 = (d0, 0) with 0 < d0 < 1 in (2.6). Fig. 4.14 depicts the
trajectory of the vortex pair, time evolution of Eε(t), Eεkin(t), xε1(t), xε2(t) and dε1(t)when ε = 1
32in NLSE and d0 = 0.5 in (2.6).
From Fig. 4.14 and additional numerical results not shown here for brevity,
we can draw the following conclusions for the interaction of vortex pair under the
NLSE dynamics with homogeneous Neumann BC: (i). The total energy is conserved
4.2 Numerical results under Neumann BC 80
-1 0 1-1
0
1
x
y
0 1 2-1
1
t
x 1ε or
x 2ε
0 1 2-1
1
t
y 1ε or
y 2ε
-1 0 1-1
0
1
x
y
0 1 2-1
1
t
x 1ε or
x 2ε
0 1 2-1
1
t
y 1ε or
y 2ε
-1 0 1-1
0
1
x
y
0 0.65 1.3-1
1
t
x 1ε or
x 2ε
0 0.65 1.3-1
1
t
y 1ε or
y 2ε
0 1 20
0.09
0.18
t
d 1ε
0 1 20
0.07
0.14
t
d 1ε
ε=1/10
ε=1/20
ε=1/40
ε=1/16
ε=1/25
ε=1/32
Figure 4.15: Trajectory and time evolution of xε1(t) and xε2(t) for d0 = 0.25 (top left
two), d0 = 0.7 (top right two) and d0 = 0.1 (bottom left two) and time evolution of
dε1(t) for d0 = 0.25 and d0 = 0.7 (bottom right two) in section 4.2.3.
numerically very well during the dynamics. (ii). The two vortices move periodically
along a circle-like trajectory for all 0 < d0 < 1 and their trajectories are symmetric.
(iii). When ε → 0, the dynamics of the two vortex centers under the NLSE dynamics
converges uniformly in time to that of the reduced dynamical laws which verifies
numerically the validation of the reduced dynamical laws in this case. In fact, based
on our extensive numerical experiments, the motions of the two vortex centers from
the reduced dynamical laws agree with those from the NLSE dynamics qualitatively
when 0 < ε < 1 and quantitatively when 0 < ε ≪ 1.
4.2.3 Vortex dipole
Here we present numerical results of the interaction of vortex dipole under the
NLSE dynamics and its corresponding reduced dynamical laws, i.e., we take M = 2,
n1 = −n2 = −1, x02 = −x0
1 = (d0, 0) with different d0 and ε = 132. Fig. 4.15 depicts
the trajectory of the vortex dipole, time evolution of xε1(t), xε2(t) and d
ε1(t).
From Fig. 4.15 and additional numerical results not shown here for brevity,
we can draw the following conclusions for the interaction of vortex dipole under the
NLSE dynamics with homogeneous Neumann BC: (i). The total energy is conserved
4.2 Numerical results under Neumann BC 81
numerically very well during the dynamics. (ii). The pattern of the motion of the
two vortices depends on their initial locations. (iii). The two vortices will move
symmetrically (and periodically if they are well separated) with respect to y-axis.
Moreover, there exists a critical value drc = dεc = dc for 0 < ε < 1, which is found
numerically as dc = 0.5, such that if initially d0 < dc, the two vortices will move
firstly upwards to the top boundary, then turn outwards to the side boundary and
finally move counter-clockwise and clockwise, respectively (cf. Fig. 4.15). While if
d0 > dc, then they will move firstly downwards to the bottom boundary, then turn
inwards to the domain and finally move counter-clockwise and clockwise, respectively
(cf. Fig. 4.15). Certainly, when d0 = 0.5, the vortex dipole does not move due to
symmetry. (iv). For fixed 0 < ε < 1, there exists another critical value dεc satisfying
limε→0 dεc = 0, such that if d0 < dεc, the vortex dipole in the NLSE dynamics will
merge at a finite time Tc depending on ε and d0 (cf. Fig. 4.15). However, the vortex
dipole from the reduced dynamical laws never merges at finite time. Hence, the
reduced dynamical laws fail qualitatively if 0 < d0 < dε0. (v). For fixed d0, when
ε → 0, the dynamics of the two vortex centers in the NLSE dynamics converges
uniformly in time to that of the reduced dynamical laws before they merge (cf.
Figs. 4.15) which verifies numerically the validation of the reduced dynamical laws
in this case. In fact, based on our extensive numerical experiments, the motions of
the two vortex centers from the reduced dynamical laws agree with those from the
NLSE dynamics qualitatively when 0 < ε < 1 and quantitatively when 0 < ε ≪ 1.
4.2.4 Vortex cluster
Here we present numerical studies on the dynamics of vortex clusters in the
NLSE with homogeneous Neumann BC, i.e., we choose the initial data (1.2) as (2.6)
and study four cases:
Case I. M = 3, n1 = n2 = n3 = 1, x01 = −x0
3 = (d0, 0) and x02 = (0, 0).
Case II. M = 4, n1 = n2 = n3 = n4 = 1, x01 = −x0
2 = (d1, 0) and x03 = −x0
4 =
(d2, 0) with 0 < d1 6= d2 < 1.
4.2 Numerical results under Neumann BC 82
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-0.4 0 0.4-0.4
0
0.4
x
y
0 0.13 0.27 0.4-0.3
0
0.3
0.6
t
x
3ε(t)
y3ε(t)
x2ε(t)
y2ε(t)
0.4 0.43 0.46 0.49
-0.3
0
0.3
0.6
t
x
3ε(t)
y3ε(t)
x2ε(t)
y2ε(t)
0.49 0.81 1.13 1.45
-0.3
0
0.3
0.6
t
x
3ε(t)
y3ε(t)
x2ε(t)
yε(t)
Figure 4.16: Trajectory of the vortex xε1 (blue line), xε2 (dark dash-dot line) and xε3
(red dash line) and their corresponding time evolution for Case I during small time
(left column), intermediate time (middle column) and large time (right column) with
ε = 140
and d0 = 0.25 in section 4.2.4.
Case III. M = 4, n1 = n2 = −n3 = −n4 = 1, x01 = −x0
2 = (d1, 0) and x03 =
−x04 = (0, d2) with 0 < d1, d2 < 1.
Case IV. M = 9, n1 = · · · = n9 = 1, and the vortex centers are initially located
on a 3×3 uniform mesh points for the rectangle [−d0, d0]×[−d0, d0] with 0 < d0 < 1.
Fig. 4.16 shows trajectory and time evolution of xε1(t), xε2(t) and xε3(t) in the
NLSE dynamics for Case I. Fig. 4.17 depicts contour plots of |ψε| at different times
in NLSE dynamics for Cases II and III, and Fig. 4.18 shows contour plots of −|ψε|and slice plots of |ψ(0, y, t)| in NLSE dynamics for Case IV to show sound wave
propagation. Based on Figs. 4.16-4.18 and additional results not shown here for
brevity, we can draw the following conclusions:
(i). For Case I, the dynamics of the vortices is quite similar as those under
Dirichlet BC in section 4.1.4 and those in trapped BEC [27, 117, 145]. The middle
vortex (initially at the origin) will not move while the other two vortices rotate
4.2 Numerical results under Neumann BC 83
Figure 4.17: Contour plots of |ψε(x, t)| with ε = 116
at different times for the NLSE
dynamics of a vortex lattice for Case II with d1 = 0.6, d2 = 0.3 (top two rows) and
Case III with d1 = d2 = 0.3 (bottom two rows) in section 4.2.4.
clockwise around the origin for some time. This dynamics agrees very well with
the NLSE dynamics in the whole plane [160, 161]. After some time, the above
symmetric structure is broken due to boundary effect and numerical errors, i.e. the
middle vortex will begin to move towards one of the other two vortices and form as
a vortex pair which will rotate with each other and also with the other single vortex
for a while. Then this pair will separate and one of them will form a new vortex pair
with the single vortex, leave the other one to be a new single vortex rotating with
them. This process will be repeated tautologically like three dancers exchange their
4.2 Numerical results under Neumann BC 84
-1 0 10
0.53
1.06t=0
y
|ψε (0
,y,t)
|
-1 0 10
0.56
1.12 t=0.006
y
|ψε (0
,y,t)
|
-1 0 10
0.58
1.16t=0.014
y
|ψε (0
,y,t)
|
-1 0 10
0.58
1.16t=0.036
y
|ψε (0
,y,t)
|
-1 0 10
0.55
1.1t=0.046
y
|ψε (0
,y,t)
|
-1 0 10
0.54
1.08t=0.051
y
|ψε (0
,y,t)
-1 0 10
0.59
1.18 t=0.07
y
|ψε (0
,y,t)
-1 0 10
0.59
1.18t=0.264
y|ψ
ε (0,y
,t)
Figure 4.18: Contour plots of −|ψε(x, t)| (left) and slice plots of |ψε(0, y, t)| (right)at different times under the NLSE dynamics of a vortex cluster in Case IV with
d0 = 0.15 and ε = 140
for showing sound wave propagation in section 4.2.4.
partners alternatively. In fact, the dynamical pattern after the symmetric structure
breaking depends highly on the mesh size and time step. (ii). For Case II, the four
vortices form as two vortex pairs when t is small. These two pairs rotate with each
other clockwise, meanwhile, the two vortices in each pair also rotate with each other
clockwise, and radiations and sound waves are emitted. The sound waves propagate
radially and are reflected back into the domain when they hit the boundary, which
push the two vortex pairs get closer. When the two vortex pairs get close enough,
the two vortices with smallest distance among the four form a new vortex pair and
leave the rest two as single vortex individually. The vortex pair rotates around the
origin. This process is iteratively repeated during the dynamics (cf. Fig. 4.17 top
two rows). (iii). For Case III, when t is small, the four vortices form as two vortex
4.2 Numerical results under Neumann BC 85
dipoles and they move symmetrically with respect to the line y = −x towards the
top right and bottom left corners, respectively. Meanwhile, the two vortices in each
dipole move symmetrically with respect to the line y = x. After a while and when
the two dipoles arrive at some places near the corners, the two vortices in each
dipole split with each other and re-formulate two different dipoles. After this, the
two vortices in each dipole move symmetrically with respect to the line y = −x,and the two new dipoles then move symmetrically with respect to the line y = x
towards their initial locations. This process is then repeated periodically (cf. Fig.
4.17 bottom two rows). (iv). For Case IV, the vortex initially centered at the origin
does not move due to symmetry, and the other eight vortices rotate clockwise and
move along two circle-like trajectories (cf. Fig. 4.18). During the dynamics, sound
waves are generated and they propagate outwards. Some of the sound waves will
exit out of the domain while others are reflected back into the domain when they
hit the boundary. The distances between the one located at the origin and the other
vortices become larger when the sound waves are radiated outwards, while they
decrease when the sound waves are reflected from the boundary and move back into
the domain (cf. Fig. 4.18).
4.2.5 Radiation and sound wave
Here we study numerically how the radiation and sound waves affect the dy-
namics of quantized vortices in the NLSE dynamics under homogeneous Neumann
BC. To this end, we take the initial data (1.2) as (4.1) with ψε0 chosen as (2.6) with
M = 2, n1 = n2 = 1 and x01 = −x0
2 = (0.1, 0) and h(x) as (3.3). Then we take δ = ε
and let ε goes to 0, and solve the NLSE with initial condition (4.1) for the vortex
centers xδ,ε1 (t) and xδ,ε2 (t) and compare them with those from the reduced dynamical
law. We denote dδ,εj (t) = |xδ,εj (t)− xrj(t)| for j = 1, 2 as the error. Fig. 4.19 depicts
time evolution of dδ,ε1 (t) for the case when δ = ε, i.e., small perturbation, and the
case when δ = 0, i.e., no perturbation. From this figure, we can see that the dynam-
ics of the two vortex centers under the NLSE dynamics converge to those obtained
4.3 Conclusion 86
0 0.035 0.070
0.02
0.04
t
d 1δ,ε
δ=0, ε=1/64
δ=0, ε=1/80
δ=0, ε=1/100
0 0.035 0.070
0.006
0.012
t
d 1δ,ε
δ=ε=1/64
δ=ε=1/80
δ=ε=1/100
Figure 4.19: Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 4.2.5
from the reduced dynamical law when ε → 0 without perturbation (cf. Fig. 4.19
left). On the contrary, the two vortex centers under the NLSE dynamics do not
converge to those obtained from the reduced dynamical law when ε→ 0 with small
perturbation (cf. Fig. 4.19 right). This clearly demonstrates radiation and sound
wave effect on vortices under the NLSE dynamics with homogenous Neumann BC.
4.3 Conclusion
In this chapter, by applying the efficient and accurate numerical methods pro-
posed in chapter 2 to simulate nonlinear Schrodinger equation (NLSE) with a di-
mensionless parameter 0 < ε < 1 in bounded domains under either Dirichlet or
homogenous Neumann BC as well as its corresponding reduced dynamical laws for
the dynamics of M quantized vortex centers, we studied numerically quantized vor-
tex dynamics and interaction and investigated the sound-vortex interaction [64,119]
in NLSE for superfluidity as well as examined the validity of the corresponding re-
duced dynamical laws under different initial setups. Based on extensive numerical
results, firstly, we verified that the dynamics of vortex centers under the NLSE dy-
namics converges to that of the reduced dynamical laws when ε → 0 before they
collide and/or move too close to the boundary. Certainly the reduced dynamical law
is only valid up to the first collision time of any two vortices, therefore they cannot
show the vortex dynamics after collision, which, however, can be observed and inves-
tigated by our directly numerical simulations. Secondly, we identified the parameter
4.3 Conclusion 87
regimes where the reduced dynamical laws agree with quantitatively and/or quali-
tatively as well as fail to agree with those from the NLSE dynamics. We concluded
that the dynamical pattern of two vortices depend on the initial phase shift (in the
case of Dirichlet BC) as well as the initial distance of the two vortices. Thirdly, We
also found that the boundary effect affect the vortex interaction very much, which
lead to very different nonlinear phenomena from those observed in the whole plane
case. The Dirichlet BC might correspond to introduce a tangential force (clock-
wise force for negative charged vortex while counter-clockwise for positive charged
vortex) while the Neumann BC might correspond to apply a normal force at the
boundary to the vortices, hence vortices might move according to a crescent-like
trajectory, or move nearly parallel to the boundary without going outside the do-
main, or merge near the boundary in some situation. Moreover, we found that the
radiation of NLSE dynamics which is carried by oscillating sound waves modifies
the motion of vortices much, especially in the dynamics of vortex clusters, highly
co-rotating vortex pairs and overlapping vortices. However, it should be reminded
that the motion of the vortices still qualitatively obeys the reduced dynamical law
when sound waves have moved away from them, either absorbed by the vortex core
or the boundary. Finally, we would like to remark here that it should be an in-
teresting question to find out how the dynamics pattern of the vortices depend on
the domain shape and size as well as the distances between vortices, and it might
be fascinating and difficult problems to extend the reduced dynamical laws for the
motion of vortices involving vortex collision and splitting, which has been conducted
by Serfaty [138] and Bethuel et al. [32] in the context of Ginzburg-Landau equation,
as well as find possible corrections to the reduced dynamical laws due to radiation.
Chapter 5
Vortex dynamics in CGLE
In this chapter, we apply the numerical method presented in chapter 2 to sim-
ulate quantized vortex interaction of CGLE, i.e., λε 6= 0, β 6= 0 in GLSE (1.1),
with different ε and under different initial setups including single vortex, vortex
pair, vortex dipole and vortex cluster. Let λε = αln(1/ε)
, we study how the dimen-
sionless parameter ε, initial setup, boundary value and geometry of the domain Daffect the dynamics and interaction of vortices. Moreover, we compare the results
obtained from the CGLE with those from the corresponding reduced dynamical
laws, and identify the cases where the reduced dynamical laws agree qualitatively
and/or quantitatively as well as fail to agree with those from CGLE on vortex in-
teraction. Finally, we also obtain numerically different patterns of the steady states
for quantized vortex clusters and study the alignment of the vortices in the steady
state.
Without specification, we let α = β = 1 and choose the initial data as the same
one in section 3.1 in chapter 3.
88
5.1 Numerical results under Dirichlet BC 89
5.1 Numerical results under Dirichlet BC
5.1.1 Single vortex
In this subsection, we present numerical results of the motion of a single quan-
tized vortex in the CGLE dynamics and the corresponding reduced dynamics, i.e
we take M = 1, n1 = 1 in (2.6). To study how the initial phase shift h(x) and
initial location of the vortex x0 affect the motion of the vortex and to understand
the validity of the reduced dynamical law, we consider the following 12 cases:
• Case I-III: x01 = (0, 0), and h(x) is chosen as Mode 1, 2 and 3, respectively;
• Case IV-VIII: x01 = (0.1, 0), and h(x) is chosen as Mode 1, 2, 3, 4 and 5,
respectively;
• Case IX-XII: x01 = (0.1, 0.2), and h(x) is chosen as Mode 2, 3, 4 and 5, respec-
tively.
Moreover, to study the effect of domain geometry, we consider domains D of three
types: type I: D = [−1, 1] × [−1, 1], type II: D = [−1, 1] × [−0.65, 0.65], type III:
D = B1(0), and study additionally the following 4 cases:
• Case XIII-XIV: x01 = (0, 0), h(x) = x + y and domain D is chosen as type II
and III, respectively;
• Case XV-XVI: x01 = (0.1, 0.2), h(x) = x2− y2 and domain D is chosen as type
II and III, respectively.
Fig. 5.1 depicts trajectory of the vortex center when ε = 132
for cases II-IV and VI
as well as time evolution of dε1(t) for different ε for cases II and VI. Fig. 5.2 depicts
trajectory of the vortex center for cases V-XII, while Fig. 5.3 shows that for cases
I, X, XIII-XVII when ε = 132
in CGLE. From Figs. 5.1-5.3 and additional numerical
experiments not shown here for brevity, we can draw following conclusions: (i).
When h(x) ≡ 0, the vortex center doesn’t move, which is similar to the vortex
5.1 Numerical results under Dirichlet BC 90
(a)-1 0 1
-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
(b)-1 0 1
-1
0
1
x
y
0 0.75 1.50
0.07
0.14
t
d 1ε
ε=1/16
ε=1/32
ε=1/64
0 1 20
0.12
0.24
t
d 1ε
ε=1/16
ε=1/32
ε=1/64
Figure 5.1: Trajectory of the vortex center in CGLE under Dirichlet BC when
ε = 132
for cases II-IV and VI and time evolution of dε1 for different ε for cases II and
VI (from left to right and then from top to bottom) in section 5.1.1.
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
Case VCase VICase VIICase VIII
Case VIIIICase XCase XICase XII
Figure 5.2: Trajectory of the vortex center in CGLE under Dirichlet BC when ε = 132
for cases IV-VII (left) and cases V-XII (right) in section 5.1.1.
dynamics in the whole space and in GLE and NLSE dynamics. (ii). When h(x) =
(x+by)(x− yb) with b 6= 0, the vortex does not move if x0 = (0, 0), while it does move
if x0 6= (0, 0) (cf. case III and VI for b = 1). This is also same with the phenomena
in GLE and NLSE dynamics. (iii). When h(x) 6= 0 and h(x) 6= (x + by)(x − yb)
with b 6= 0, in general, the vortex center does move to a different point from its
5.1 Numerical results under Dirichlet BC 91
(a)-1 0 1
-1
0
1
x
y
-1 0 1-0.65
0
0.65
x
y
(b)-1 0 1
-1
0
1
x
y
-1 0 1-0.65
0
0.65
x
y
Figure 5.3: Trajectory of the vortex center in CGLE under Dirichlet BC when ε = 132
for cases: (a) I, XIII, XIV, (b) X, XV, XVI (from left to right) in section 5.1.1.
initial location and stays there forever. This is quite different from the situation
in the whole space, where a single vortex may move to infinity under the initial
data (2.6) with h(x) 6= 0. (iv). In general, the initial location, the geometry of the
domain and the boundary value will all affect the motion of the vortex center. (v).
When ε → 0, the dynamics of the vortex center in the CGLE dynamics converges
uniformly in time to that in the reduced dynamics (cf. Fig. 5.1) which verifies
numerically the validation of the reduced dynamical laws. In fact, based on our
extensive numerical experiments, the motion of the vortex center from the reduced
dynamical laws agrees with those from the CGLE dynamics qualitatively when 0 <
ε < 1 and quantitatively when 0 < ε≪ 1.
5.1.2 Vortex pair
Here we present numerical results of the interaction of vortex pair in the CGLE
dynamics and its corresponding reduced dynamics, i.e., we takeM = 2, n1 = n2 = 1,
x01 = (−0.3, 0) and x0
2 = (0.3, 0) in (2.6). Fig. 5.4 depicts the trajectory of the vortex
5.1 Numerical results under Dirichlet BC 92
(a) -1 0 1-1
0
1
x
y
Mode 0Mode 1Mode 2
0 3.2 6.420
32
t
Eε(t)
0 3.2 6.417
29
t
Eε kin(t)
0 3.2 6.43.1
3.25
3.4
t
Eε in
t(t)
(b) -1 0 1-1
0
1
x
y
Mode 3Mode 4Mode 5
0 1.6 3.216
40
t
Eε(t)
0 1.6 3.214
36
t
Eε kin(t)
0 1.6 3.22.8
3.2
3.6
t
Eε in
t(t)
Figure 5.4: Trajectory of the vortex centers (a) and their corresponding time
evolution of the GL functionals (b) in CGLE dynamics under Dirichlet BC when
ε = 125
with different h(x) in (2.6) in section 5.1.2.
centers and their corresponding time evolution of the GL functionals when ε = 125
in the CGLE with different h(x) in (2.6), while Fig. 5.5 shows contour plots of
|ψε(x, t)| for ε = 125
at different times as well as the time evolution of xε1(t), xr1(t)
and dε1(t) for different ε with h(x) = 0 in (2.6).
From Figs. 5.4 and 5.5 and additional numerical experiments now shown here
for brevity, we can draw the following conclusions for the interaction of vortex pair
in the CGLE dynamics with Dirichlet BC: (i). The two vortices undergo a repulsive
interaction, they never collide. They rotate with each other and meanwhile move
apart from each other towards the boundary of D and finally stop somewhere near
the boundary, which indicates that the boundary imposes a repulsive force on the
vortices (cf. Fig. 5.4). As shown in previous chapters, a vortex pair in the GLE
dynamics moves outward along the line that connects the two vortices and finally
stay steady near the boundary, while in the NLSE dynamics the two vortices always
rotate around each other periodically. Hence, the motion of the vortex pair here is
5.1 Numerical results under Dirichlet BC 93
somehow the combination of that in the GLE and NLSE dynamics. Actually, from
extensive numerical results, we find that the larger the value β (α) is, the closer
the motion in CGLE dynamics is to that in NLSE (GLE) dynamics, which evidence
numerically that the CGLE under Dirichlet BC is somehow in between the GLE
and NLSE under Dirichlet BC. (ii). The phase shift h(x) affects the motion of the
vortices significantly. When h(x) = (x + by)(x − yb) with b 6= 0, the vortices will
move outward symmetric with respect to the origin, i.e., x1(t) = −x2(t) (cf. Fig.
5.4). (iii). When ε → 0, the dynamics of the two vortex centers in the CGLE
dynamics converges uniformly in time to that in the reduced dynamics (cf. Fig.
5.5) which verifies numerically the validation of the reduced dynamical laws in this
case. In fact, based on our extensive numerical experiments, the motions of the two
vortex centers from the reduced dynamical laws agree with those from the CGLE
dynamics qualitatively when 0 < ε < 1 and quantitatively when 0 < ε ≪ 1. (iv).
During the dynamics of CGLE, the GL functional and its kinetic part decrease
when time increases, its interaction part changes dramatically when t is small, and
when t → ∞, all the three quantities converge to constants (cf. Fig. 5.4), which
immediately imply that a steady state solution will be reached when t→ ∞.
5.1.3 Vortex dipole
Here we present numerical results of the interaction of vortex dipole under the
CGLE dynamics and its corresponding reduced dynamical laws, i.e., we takeM = 2,
n1 = −n2 = −1, x02 = −x0
1 = (0.3, 0) in (2.6). Fig. 5.6 depicts the trajectory of the
vortex centers and their corresponding time evolution of the GL functionals when
ε = 125
in the CGLE with different h(x) in (2.6), while Fig. 5.7 shows contour
plot of |ψε(x, t)| for ε = 125
at different times as well as the time evolution of xε1(t),
xr1(t) and dε1(t) for different ε with h(x) = 0 in (2.6). From Figs. 5.6 and 5.7
and additional numerical experiments now shown here for brevity, we can draw the
following conclusions for the interaction of vortex dipole in the CGLE dynamics
with Dirichlet BC: (i). The two vortices undergo an attractive interaction, they
5.1 Numerical results under Dirichlet BC 94
0 3 6-0.75
-0.53
-0.31
t
x 1ε (t) o
r x1r (t)
ε=1/16
ε=1/40reduced
0 3 60
0.3
0.6
t
y 1ε (t) o
r y1r (t)
ε=1/16ε=1/40reduced
0 3 60
0.08
0.16
t
d 1ε (t)
ε=1/16ε=1/25ε=1/40
Figure 5.5: Contour plot of |ψε(x, t)| for ε = 125
at different times as well as time
evolution of xε1(t) in CGLE dynamics and xr1(t) in the reduced dynamics under
Dirichlet BC with h(x) = 0 in (2.6) and their difference dε1(t) for different ε in
section 5.1.2.
will collide and annihilation with each other. (ii). The phase shift h(x) and the
initial distance of the two vortices affect the motion of the vortices significantly. If
h(x) = 0, the vortex dipole will finally merge regardless where they are initially
located. However, similar as the case in GLE dynamics, if h(x) 6≡ 0, say h(x) =
x + y for example, there would be a critical distance dεc, which depend on the
value of ε, that divide the motion of the vortex dipole into two groups: (a) if
the distance between the vortex dipole initially |x02 − x0
1| > dεc, the vortex will
never merge, they will finally stay steady and separately at some place that near
the boundary. (b) otherwise, they do finally merge and annihilation. (iii). For
h(x) = 0, when ε → 0, the dynamics of the two vortex centers in the CGLE
dynamics converges uniformly in time to that in the reduced dynamics (cf. Fig.
5.7) which verifies numerically the validation of the reduced dynamical laws in this
case. In fact, based on our extensive numerical experiments, the motions of the two
5.1 Numerical results under Dirichlet BC 95
(a) -0.4 0 0.4-0.1
0.3
0.7
x
y
Mode 0Mode 1Mode 2
0 0.18 0.360
28
t
Eε(t)
0 0.18 0.360
25
t
Eε kin(t)
0 0.18 0.360
1.8
3.6
t
Eε in
t(t)
(b) -0.35 0 0.35-0.1
0.25
0.6
x
y
Mode 3Mode 4Mode 5
0 0.12 0.242
32
t
Eε(t)
0 0.12 0.242
28
t
Eε kin(t)
0 0.12 0.240
1.8
3.6
t
Eε in
t(t)
Figure 5.6: Trajectory of the vortex centers (a) and their corresponding time
evolution of the GL functionals (b) in CGLE dynamics under Dirichlet BC when
ε = 125
with different h(x) in (2.6) in section 5.1.3.
vortex centers from the reduced dynamical laws agree with those from the CGLE
dynamics qualitatively when 0 < ε < 1 and quantitatively when 0 < ε ≪ 1 before
they merge. (iv). During the dynamics of CGLE, the GL functional decreases when
time increases, its kinetic and interaction parts don’t change dramatically when t is
small, and when t → ∞, all the three quantities converge to constants. Moreover,
if finite time merging/annihilation happens, the GL functional and its kinetic and
interaction parts change significantly during the collision. In addition, when t→ ∞,
the interaction energy goes to 0 which immediately implies that a steady state will
be reached in the form of φε(x) = eic(x), where c(x) is a harmonic function satisfying
c(x)|∂D = h(x) +∑M
j=1 njθ(x− x0j ).
5.1.4 Vortex cluster
Here we present numerical results of the interaction of vortex clusters under the
CGLE dynamics. We consider the following cases: case I.M = 3, n1 = n2 = n3 = 1,
5.1 Numerical results under Dirichlet BC 96
0 0.046 0.0920
0.14
0.28
t
y 1ε (t) o
r y1r (t)
ε=1/25
ε=1/64reduced
0 0.046 0.092-0.3
-0.17
-0.04
t
x 1ε (t) o
r x1r (t)
ε=1/25
ε=1/64reduced
0 0.048 0.0960
0.05
0.1
t
d 1ε (t)
ε=1/25
ε=1/32
ε=1/64
Figure 5.7: Contour plots of |ψε(x, t)| for ε = 125
at different times as well as time
evolution of xε1(t) in CGLE dynamics, xr1(t) in the reduced dynamics under Dirichlet
BC with h(x) = 0 in (2.6) and their difference dε1(t) for different ε in section 5.1.3.
x01 = (0.5, 0); x0
2 = (−0.25,√34), x0
3 = (−0.25,−√34), case II. M = 3, n1 = n2 = n3 =
1, x01 = (−0.4, 0), x0
2 = (0, 0), x03 = (0.4, 0); case III. M = 3, n1 = n2 = n3 = 1,
x01 = (0, 0.3), x0
2 = (0.15, 0.15), x03 = (0.3, 0); case IV. M = 3, −n1 = n2 = n3 = 1,
x01 = (0.5, 0); x0
2 = (−0.25,√34), x0
3 = (−0.25,−√34), case V. M = 3, n2 = −1,
n1 = n3 = 1, x01 = (−0.4, 0), x0
2 = (0, 0), x03 = (0.4, 0); case VI. M = 3, n1 = −1,
n2 = n3 = 1, x01 = (0.2, 0.3), x0
2 = (−0.3, 0.4), x03 = (−0.4,−0.2); case VII. M = 4,
n1 = n2 = n3 = n4 = 1, x01 = (0.5, 0), x0
2 = (0, 0.5), x03 = (−0.5, 0), x0
4 = (0,−0.5);
case VIII. M = 4, n1 = n3 = 1, n2 = n4 = −1, x01 = (0.5, 0), x0
2 = (0, 0.5),
x03 = (−0.5, 0), x0
4 = (0,−0.5); case IX. M = 4, n2 = n3 = −1, n1 = n4 = 1, x01 =
(0.5, 0), x02 = (0, 0.5), x0
3 = (−0.5, 0), x04 = (0,−0.5); case X. M = 4, n1 = n3 = 1,
n2 = n4 = −1, x01 = (0.5, 0.5), x0
2 = (−0.5, 0.5), x03 = (−0.5,−0.5), x0
4 = (0.5,−0.5);
case XI. M = 4, n2 = n3 = −1, n1 = n4 = 1, x01 = (0.5, 0.5), x0
2 = (−0.5, 0.5),
x03 = (−0.5,−0.5), x0
4 = (0.5,−0.5); case XII. M = 4, n1 = n3 = −1, n2 = n4 = 1,
x01 = (0.4, 0), x0
2 = (−0.4/3, 0), x03 = (0.4/3, 0), x0
4 = (0.4, 0); case XIII. M = 4,
5.1 Numerical results under Dirichlet BC 97
n2 = n3 = −1, n1 = n4 = 1, x01 = (0.4, 0), x0
2 = (−0.4/3, 0), x03 = (0.4/3, 0),
x04 = (0.4, 0); case XIV. M = 4, n1 = n2 = −1, n3 = n4 = 1, x0
1 = (0.4, 0),
x02 = (−0.4/3, 0), x0
3 = (0.4/3, 0), x04 = (0.4, 0); case XV. M = 4, n1 = n3 = −1,
n2 = n4 = 1, x01 = (0.2, 0.3), x0
2 = (−0.3, 0.4), x03 = (−0.4,−0.2); x0
4 = (0.3,−0.3);
Fig. 5.8 shows trajectory of the vortex centers when ε = 132
in 1.1 and h(x) = 0 in
(2.6) for the above 15 cases. From Fig. 5.8 and additional numerical experiments not
shown here for brevity, we can draw the following conclusions: (i). The interaction
of vortex clusters under the CGLE dynamics with Dirichlet BC is very interesting
and complicated. The detailed dynamics and interaction pattern of a vortex cluster
depends on its initial alignment of the vortices, geometry of the domain D and the
boundary value g(x). (ii). For a cluster of M vortices, if they have the same index,
then no collision will happen for any time t ≥ 0. On the other hand, if they have
opposite index, e.g. M+ > 0 vortices with index ‘+1’ and M− > 0 vortices with
index ‘−1’ satisfying M+ +M− = M , collision will always happen at finite time.
In addition, when t is sufficiently large, there exist exactly |M+ −M−| vortices of
winding number ‘+1’ ifM+ > M−, and resp. ‘−1’ ifM+ < M−, left in the domain.
5.1.5 Steady state patterns of vortex clusters
Here we present the steady state patterns of vortex clusters in the CGLE dynam-
ics under Dirichlet BC. We study how the geometry of the domain D and boundary
condition affect the alignment of vortices in the steady states. To this end, we take
ε = 132
in,
nj = 1, x0j = 0.5
(cos
(2jπ
M
), sin
(2jπ
M
)), j = 1, 2, . . . ,M,
i.e., initially we have M like vortices which are located uniformly in a circle centered
at origin with radius R1 = 0.5.
Denote φε(x) as the steady state, i.e., φε(x) = limt→∞ ψε(x, t) for x ∈ D. Fig.
5.9 depicts the contour plots of the amplitude |φε| of the steady state in the CGLE
dynamics with h(x) = 0 in (2.6) for differentM and domains, while Fig. 5.10 depicts
5.1 Numerical results under Dirichlet BC 98
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
Figure 5.8: Trajectory of vortex centers for the interaction of different vortex
lattices in GLE under Dirichlet BC with ε = 132
and h(x) = 0 for cases I-IX (from
left to right and then from top to bottom) in section 5.1.4.
5.1 Numerical results under Dirichlet BC 99
(a)
(b)
(c)
Figure 5.9: Contour plots of |φε(x)| for the steady states of vortex cluster in CGLE
under Dirichlet BC with ε = 132
for M = 8, 12, 16, 20 (from left column to
right column) and different domains: (a) unit disk D = B1(0), (b) square domain
D = [−1, 1]2, (c) rectangular domain D = [−1.6, 1.6]× [−0.8, 0.8].
similar results with M = 12 for different h(x) in (2.6).
From Figs. 5.9 & 5.10 and additional numerical results not shown here for
brevity, we can draw the following conclusions for the steady state patterns of vor-
tex clusters under the CGLE dynamics with Dirichlet BC: (i). The vortex undergo
repulsive interaction between each other and they move to locations near the bound-
ary of D, there is no collision and a steady state pattern is formed when t→ ∞. In
fact, the steady state is also the solution of the following minimization problem
φε = argminφ(x)|x∈∂D=ψε
0(x)|x∈∂DEε(φ).
Actually, based on our extensive numerical experiments, we found that for a vortex
cluster of any configuration, i.e., vortices in the vortex cluster may be opposite
winding number, the vortices either merge and annihilate and all the leftover vortices
5.1 Numerical results under Dirichlet BC 100
Figure 5.10: Contour plots of |φε(x)| for the steady states of vortex cluster in CGLE
under Dirichlet BC with ε = 132
and M = 12 on a unit disk D = B1(0) (top row)
or a square D = [−1, 1]2 (middle row) or a rectangular domain D = [−1.6, 1.6] ×[−0.8, 0.8] (bottom row) under different h(x) = x + y, x2 − y2, x − y, x2 − y2 +
2xy, x2 − y2 − 2xy (from left column to right column).
are all pinned in near the boundary finally. This phenomena is similar with the one
in the superconductor involving magnetic field [104]. (ii). During the dynamics, the
GL functional decreases when time increases. (iii). Both the geometry of the domain
and the boundary condition, i.e., h(x), affect the final steady states significantly.
(iv). At the steady state, the distance between the vortex centers and ∂D depends on
ε andM . For fixedM , when ε decreases, the distance decreases; and respectively, for
fixed ε, whenM increases, the distance decreases. We remark it here as a interesting
open problem to find how the width depend on the value of ε, the boundary condition
as well as the geometry of the domain.
5.1 Numerical results under Dirichlet BC 101
0 3 60
0.13
0.26
t
d1ε (t
)
0 3 60
0.08
0.16
t
d1ε (t
)
δ=0,ε=1/16
δ=0,ε=1/25
δ=0,ε=1/40
δ=ε=1/16δ=ε=1/32δ=ε=1/50
Figure 5.11: Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 5.1.6
5.1.6 Validity of RDL under small perturbation
As seen from former chapters, the small perturbation affects the motion of the
vortices in NLSE dynamics much but hardly affects those in the GLE dynamics.
The question that how this affects those vortex motion in CGLE dynamics now
come up naturally. To this end, similar as the one studied in the GLE dynamics, we
take the initial data (1.2) as (3.4) with ψε0 chosen as (2.6) with h(x) ≡ 0, M = 2,
n1 = n2 = 1 and x01 = −x0
2 = (0.3, 0). Then we take δ = ε and let ε go to 0, and
solve the CGLE under Dirichlet BC and with initial condition (3.4) for the vortex
centers xδ,ε1 (t) and xδ,ε2 (t) and compare them with those from the reduced dynamical
law. We denote dδ,εj (t) = |xδ,εj (t)− xrj(t)| for j = 1, 2 as the error. Fig. 5.11 depicts
time evolution of dδ,ε1 (t) for the case when δ = ε, i.e., small perturbation, and the
case when δ = 0, i.e., no perturbation. From this figure, we can see that small
perturbation in the initial data does not affect the motion of the vortices much,
same as the non-perturbed initial setups, the dynamics of the two vortex centers
under the CGLE dynamics with perturbed initial setups also converge to those
obtained from the reduced dynamical law when ε → 0. Same as the case in GLE
dynamics, this situation is foreseeable since the CGLE is also a dissipative system,
and small perturbation initially imposed for a dissipative system will not affect the
system much for the dynamics.
5.2 Numerical results under Neumann BC 102
(a) -1 0 1-1
0
1
x
y
0 1.25 2.5-0.62
0.32
t
x 1ε (t)
ε=1/8
ε=1/16
ε=1/50reduced
0 1.25 2.5-1
0
t
y 1ε (t)
0 1 20
0.24
0.48
t
d 1ε (t)
ε=1/16
ε=1/25
ε=1/50
(b) -1 0 1-1
0
1
x
y
0 0.73 1.460.1
1
t
x 1ε (t)
ε=1/8
ε=1/16
ε=1/50reduced
0 0.73 1.46-0.4
0.3
t
y 1ε (t)
0 0.62 1.240
0.24
0.48
t
d 1ε (t)
ε=1/16
ε=1/25
ε=1/50
Figure 5.12: Trajectory of the vortex center when ε = 125
(left) as well as time
evolution of xε1 (middle) and dε1 for different ε (right) for the motion of a single
vortex in CGLE under homogeneous Neumann BC with different x01 in (2.6) in
section 5.2.1.: (a) x01 = (0.1, 0), (b) x0
1 = (0.1, 0.2).
5.2 Numerical results under Neumann BC
5.2.1 Single vortex
Here we present numerical results of the motion of a single quantized vortex
under the CGLE dynamics and its corresponding reduced dynamical laws, i.e., we
take M = 1 and n1 = 1 in (2.6). Fig. 5.12 depicts trajectory of the vortex center for
different x01 in (2.6) when ε = 1
25as well as time evolution of xε1 and dε1 for different
ε. From Fig. 5.12 and additional numerical results not shown here for brevity, we
can see that:
(i). The initial location of the vortex, i.e., value of x0 affects the motion of the
vortex a lot and this shows the effect on the vortex from the Neumann BC.
(ii). If x01 = (0, 0), the vortex will not move all the time, otherwise, the vortex
will move and finally exit the domain and never come back. This is quite different
5.2 Numerical results under Neumann BC 103
(a) (c)
0 0.4 0.80
14
28
t
E ε(t )
E εk i n
0 0.4 0.80
1.7
3.4
t
E εi n t
(b) (d)
0 0.17 0.340
10
20
t
E ε(t )
E εk i n
0 0.17 0.340
1.7
3.4
t
E εi nt
Figure 5.13: Contour plots of |ψε(x, t)| at different times when ε = 125
((a) &
(b)) and the corresponding time evolution of the GL functionals ((c) & (d)) for the
motion of vortex pair in CGLE under homogeneous Neumann BC with different d0
in (2.6) in section 5.2.2: top row: d0 = 0.3, bottom row: d0 = 0.7.
from the situations in bounded domain with Dirichlet BC where a single vortex can
never move outside the domain or in the whole space where a single vortex doesn’t
move at all under the initial condition (2.6) when D = R2.
(iii). As ε → 0, the dynamics of the vortex center under the CGLE dynamics
converges uniformly in time to that of the reduced dynamical laws well before it exits
the domain, which verifies numerically the validation of the reduced dynamical laws
in this case. Surely, when the vortex center is being exited the domain or after it
moves out of the domain, the reduced dynamics laws are no longer valid. However,
the dynamics of CGLE is still physically interesting. In fact, based on our extensive
numerical experiments, the motion of the vortex center from the reduced dynamical
laws agrees with that from the CGLE dynamics qualitatively when 0 < ε < 1 and
quantitatively when 0 < ε≪ 1 well before it moves out of the domain.
5.2 Numerical results under Neumann BC 104
(a) -1 0 1-1
0
1
x
y
0 0.285 0.570
0.55
t
x 1ε (t)
ε=1/16
ε=1/25
ε=1/40reduced
0 0.285 0.57-1
0
t
y 1ε (t)
0 0.275 0.550
0.15
0.3
t
d 1ε (t)
ε=1/16
ε=1/25
ε=1/40
(b) -1 0 1-1
0
1
x
y
0 0.045 0.090.7
1
t
x 1ε (t)
ε=1/16
ε=1/25
ε=1/40reduced
0 0.045 0.09-0.4
0
t
y 1ε (t)
0 0.045 0.090
0.07
0.14
t
d 1ε (t)
ε=1/16
ε=1/25
ε=1/40
Figure 5.14: Trajectory of the vortex center when ε = 125
(left) as well as time
evolution of xε1 (middle) and dε1 for different ε (right) for the motion of vortex pair
in CGLE under homogeneous Neumann BC with different d0 in (2.6) in section 5.2.2:
(a) d0 = 0.3, (b) d0 = 0.7.
5.2.2 Vortex pair
Here we present numerical results of the interaction of vortex pair under the
CGLE dynamics and its corresponding reduced dynamical laws, i.e., we takeM = 2,
n1 = n2 = 1 and x02 = −x0
1 = (d0, 0) with 0 < d0 < 1 in (2.6). Fig. 5.13 shows the
contour plots of |ψε(x, t)| at different times when ε = 125, while Fig. 5.14 depicts
the trajectory of the vortex pair when ε = 125
as well as time evolution of xε1(t) and
dε1(t) for different d0 in (2.6).
From Figs. 5.14 and 5.13 and additional numerical results not shown here for
brevity, we can draw the following conclusions for the interaction of vortex pair
under the NLSE dynamics with homogeneous Neumann BC:
(i). The initial location of the vortex, i.e., value of d0 affects the motion of the
vortex a lot and this shows the effect on the vortex from the Neumann BC.
(ii). For the CGLE with ε fixed, there exist a sequence of critical values dc,ε1 >
5.2 Numerical results under Neumann BC 105
dc,ε2 > dc,ε3 > · · · > dc,εk > · · · such that: if d0 > dc,ε1 , the two vortex will exit the
domain from the side boundary; if dc,ε1 > d0 > dc,ε2 , the two vortex will exit the
domain from the top-bottom boundary; if dc,ε2 > d0 > dc,ε3 , the two vortex will exit
the domain from the side boundary again. Actually, they exit from the side or top-
bottom boundary alternatively, i.e., for n = 0, 1, · · · : if dc,ε2n > d0 > dc,ε2n+1, the two
vortex will exit the domain from the side boundary; otherwise, if dc,ε2n+1 > d0 > dc,ε2n+2,
the two vortex will exit the domain from the top-bottom boundary. For the reduced
dynamical law, there also exist such series of critical values dc,rk , k = 0, · · · which
divide the patterns of the trajectory. It might be an interesting problem to find
those dc,εk and dc,rk , and to study how they agree with each other.
Again, the motion here is somehow the combination of that in the GLE dynam-
ics and that in the NLSE dynamics. The vortex pair in the GLE dynamics will
always move outward along the line that connects the two vortices and finally exit
the domain, while in the NLSE dynamics, they always rotate around each other
periodically. Actually, from extensive numerical results, we find that for a fixed
initial setups, the larger the value β is, the more rotation the pair will do before
they exit the domain, i.e, the closer the motion in CGLE dynamics is to that in
NLSE dynamics; on contrary, the larger the value α, the faster the vortex exit the
domain, in other words, the closer the motion in CGLE dynamics is to that in NLSE
dynamics. This again evidence numerically that the CGLE under Neumann BC is
somehow in between the GLE and NLSE under Neumann BC.
(iii). As ε → 0, the dynamics of the two vortex centers under the CGLE dynamics
converge uniformly in time to that of the reduced dynamical laws well before any
one of them exit the domain, which verifies numerically the validation of the reduced
dynamical laws in this case. Surely, when the vortex centers are being exiting the
domain or after they moves out of the domain, the reduced dynamics laws are no
longer valid. However, the dynamics of CGLE is still physically interesting. In
fact, based on our extensive numerical experiments, the motions of the two vortex
centers from the reduced dynamical laws agree with those from the CGLE dynamics
5.2 Numerical results under Neumann BC 106
(a) (c)
0 0.3 0.60
10
20
t
E ε(t )E εk i n
0 0.3 0.60
1.7
3.4
t
E εi n t
(b) (d)
0 0.3 0.60
10
20
t
E ε(t )E εk i n
0 0.3 0.60
1.7
3.4
t
E εi n t
Figure 5.15: Contour plots of |ψε(x, t)| at different times when ε = 125
and the
corresponding time evolution of the GL functionals for the motion of vortex dipole
in CGLE under homogeneous Neumann BC with different d0 in (2.6) in section 5.2.3:
top row: d0 = 0.3, bottom row: d0 = 0.7.
qualitatively when 0 < ε < 1 and quantitatively when 0 < ε≪ 1.
(iv). During the dynamics of CGLE, the GL functional and its kinetic parts
decrease when time increases. They doesn’t change much when t is small and changes
dramatically when either one of the two vortices moves outside the domain D. When
t→ ∞, all the three quantities converge to 0 (cf. Fig. 5.13 (c) & (d)), which imply
that a constant steady state will be reached in the form of φε(x) = eic0 for x ∈ Dwith c
0a constant.
5.2.3 Vortex dipole
Here we present numerical results of the interaction of vortex dipole in the CGLE
dynamics and its corresponding reduced dynamics, i.e., we takeM = 2, n2 = −n1 =
1 and x02 = −x0
1 = (d0, 0) with 0 < d0 < 1 in (2.6).
Fig. 5.15 shows the contour plots of |ψε(x, t)| at different times when ε = 125,
while Fig. 5.16 depicts the trajectory of the vortex pair when ε = 125
as well as time
5.2 Numerical results under Neumann BC 107
-1 0 1-1
0
1
x
y
0 0.07 0.140
0.3
t
x 1ε (t)
ε=1/25
ε=1/32
ε=1/50reduced
0 0.07 0.140
0.46
t
y 1ε (t)
0 0.054 0.1080
0.06
0.12
t
d 1ε (t)
ε=1/25
ε=1/32
ε=1/50
-1 0 1-1
0
1
x
y
0 0.07 0.140.7
1
t
x 1ε (t)
ε=1/25
ε=1/32
ε=1/50reduced
0 0.07 0.14-0.4
0
t
y 1ε (t)
0 0.054 0.1080
0.07
0.14
t
d 1ε (t)
ε=1/25
ε=1/32
ε=1/50
Figure 5.16: Trajectory of the vortex center when ε = 125
(left) as well as time
evolution of xε1 (middle) and dε1 for different ε (right) for the motion of vortex dipole
in CGLE under homogeneous Neumann BC with different d0 in (2.6) in section 5.2.2:
(a) d0 = 0.3, (b) d0 = 0.7.
evolution of xε1(t) and dε1(t) for different d0 in (2.6).
From Fig. 5.16 and 5.15 and additional numerical results not shown here for
brevity, we can draw the following conclusions for the interaction of vortex pair
under the NLSE dynamics with homogeneous Neumann BC:
(i). The initial location of the vortex, i.e., value of d0 affects the motion of the
vortex a lot and this shows the effect on the vortex from the Neumann BC. (ii).
For the CGLE with ε fixed, there exists a critical value dεc such that: if d0 > dεc,
the two vortices will exit the domain from the side boundary, otherwise, they will
merge somewhere in the boundary. For the reduced dynamical law, there also exists
such series of critical values drc which divide the patterns of the trajectory. It might
be an interesting problem to find those dεc and drc, and to study how they agree
with each other. Moreover, the trajectories of the two vortices are symmetric i.e.,
x1(t) = −x2(t), and finally the CGLE dynamics will lead to a constant steady state
5.2 Numerical results under Neumann BC 108
with amplitude 1, i.e., φε(x) = eic0 for x ∈ D with c0 a real constant. (iii). As
ε→ 0, the dynamics of the two vortex centers under the CGLE dynamics converge
uniformly in time to that of the reduced dynamical laws well before they move out
of the domain or merge with each other, which verifies numerically the validation of
the reduced dynamical laws in this case. In fact, based on our extensive numerical
experiments, the motions of the two vortex centers from the reduced dynamical
laws agree with those from the CGLE dynamics qualitatively when 0 < ε < 1 and
quantitatively when 0 < ε≪ 1.
5.2.4 Vortex cluster
Here we present numerical results of the interaction of vortex clusters under the
CGLE dynamics. We consider the following 15 cases:
case I. M = 3, n1 = n2 = n3 = 1, x01 = (0.4, 0); x0
2 = (−0.2,√35), x0
3 =
(−0.2,−√35), case II. M = 3, n1 = n2 = n3 = 1, x0
1 = (−0.4, 0.2), x02 = (0, 0.2),
x03 = (0.4, 0.2); case III. M = 3, n1 = n2 = n3 = 1, x0
1 = (−0.4, 0), x02 = (0, 0),
x03 = (0.4, 0); case IV. M = 3, −n1 = n2 = n3 = 1, x0
1 = (0.4, 0); x02 = (−0.2,
√35),
x03 = (−0.2,−
√35), case V. M = 3, −n2 = n1 = n3 = 1, x0
1 = (−0.4, 0), x02 = (0, 0),
x03 = (0.4, 0); case VI. M = 3, −n2 = n1 = n3 = 1, x0
1 = (−0.7, 0), x02 = (0, 0),
x03 = (0.7, 0); case VII. M = 4, n1 = n2 = n3 = n4 = 1, x0
1 = (0.4, 0), x02 = (0, 0.4),
x03 = (−0.4, 0), x0
4 = (0,−0.4); case VIII. M = 4, n1 = n3 = −1, n2 = n4 = 1,
x01 = (0.4, 0), x0
2 = (0, 0.4), x03 = (−0.4, 0), x0
4 = (0,−0.4); case IX. M = 4,
n1 = n3 = −1, n2 = n4 = 1, x01 = (0.59, 0), x0
2 = (0, 0.59), x03 = (−0.59, 0),
x04 = (0,−0.59); case X. M = 4, n1 = n3 = −1, n2 = n4 = 1, x0
1 = (0.7, 0),
x02 = (0, 0.7), x0
3 = (−0.7, 0), x04 = (0,−0.7); case XI. M = 4, n2 = n3 = −1,
n1 = n4 = 1, x01 = (0.4, 0), x0
2 = (0, 0.4), x03 = (−0.4, 0), x0
4 = (0,−0.4); case XII.
M = 4, n2 = n3 = −1, n1 = n4 = 1, x01 = (0.6, 0), x0
2 = (0, 0.6), x03 = (−0.6, 0),
x04 = (0,−0.6); case XIII. M = 4, n1 = n3 = −1, n2 = n4 = 1, x0
1 = (0.4, 0),
x02 = (−0.4/3, 0), x0
3 = (0.4/3, 0), x04 = (0.4, 0); case XIV. M = 4, n1 = n3 = −1,
n2 = n4 = 1, x01 = (0.4, 0), x0
2 = (−0.4/3, 0), x03 = (0.4/3, 0), x0
4 = (0.4, 0); case XV.
5.2 Numerical results under Neumann BC 109
-1 0 1-1
0
1y
x -1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
-1 0 1-1
0
1
x
y
Figure 5.17: Trajectory of vortex centers for the interaction of different vortex
clusters in CGLE under Neumman BC with ε = 132
for cases I-IX (from left to right
and then from top to bottom) in section 5.2.4.
5.2 Numerical results under Neumann BC 110
M = 4, n1 = n3 = −1, n2 = n4 = 1, x01 = (−0.6, 0), x0
2 = (−0.1, 0), x03 = (0.1, 0),
x04 = (0.6, 0);
Fig. 5.17 shows trajectory of the vortex centers for the above 15 cases when
ε = 132, while Fig. 5.18 depicts the contour plots of |ψε| for the initial data and
corresponding steady states for cases I, III, V, VI, VII and XIV.
From Figs. 5.17 & 5.18 and additional numerical experiments not shown here
for brevity, we can draw the following conclusions: (i). The interaction of vortex
clusters under the CGLE dynamics with homogeneous BC is very interesting and
complicated. The detailed dynamics and interaction pattern of a cluster depends
on its initial alignment of the cluster and geometry of the domain D. (ii). For a
cluster of M vortices, if they have the same index, then at least M − 1 vortices
will move out of the domain at finite time and no collision will happen for any time
t ≥ 0. On the other hand, if they have opposite index, collision will happen at finite
time. After collisions, the leftover vortices will then move out of the domain at finite
time and at most one vortex may left in the domain. When t is sufficiently large,
in most cases, no vortex is left in the domain; of course, when the geometry and
initial setup are properly symmetric and M is odd, there maybe one vortex left in
the domain. (iii). If finally no vortex leftover in the domain, the GL functionals will
always vanish as t → ∞, which indicate that the final steady states always admit
the form of φε(x) = eic0 for x ∈ D with c0 a real constant.
5.2.5 Validity of RDL under small perturbation
Same as the motivation in section 5.1.6, here we study the radiation property of
the CGLE dynamics under homogeneous Neumann BC in this subsection.
To this end, we take the initial data (1.2) as (3.4) with ψε0 chosen as (2.6) with
M = 2, n1 = n2 = 1, x01 = −x0
2 = (0.7, 0) and h(x) as (3.3). Then we take δ = ε
and let ε go to 0, solve the CGLE with initial condition (3.4) for the vortex centers
xδ,ε1 (t) and x
δ,ε2 (t) and compare them with those from the reduced dynamical law.
We denote dδ,εj (t) = |xδ,εj (t)− xrj(t)| for j = 1, 2 as the error. Fig. 5.19 depicts time
5.2 Numerical results under Neumann BC 111
(a)
(b)
(c)
(d)
Figure 5.18: Contour plots of |ψε(x, t)| for the initial data ((a) & (c)) and cor-
responding steady states ((b) & (d)) of vortex cluster in CGLE dynamics under
Neumman BC with ε = 132
and for cases I, III, V, VI, VII and XIV (from left to
right and then from top to bottom) in section 5.2.4.
evolution of dδ,ε1 (t) for the case when δ = ε, i.e., small perturbation, and the case
when δ = 0, i.e., no perturbation. From Fig. 5.19, we can see that small perturbation
5.3 Vortex dynamics in inhomogeneous potential 112
0 0.045 0.090
0.07
0.14
t
d1ε (t
)
0 0.045 0.090
0.07
0.14
t
d1ε (t
)
δ=0,ε=1/16
δ=0,ε=1/25
δ=0,ε=1/40
δ=ε=1/16
δ=ε=1/25
δ=ε=1/40
Figure 5.19: Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and per-
turbed initial data (right) in section 5.2.5
in the initial data does not affect the motion of the vortices much, same as the non-
perturbed initial setups, the dynamics of the two vortex centers under the CGLE
dynamics with perturbed initial setups also converge to those obtained from the
reduced dynamical law when ε → 0, which is simply similar as the situation in the
CGLE dynamics with perturbed initial data under Dirichlet BC.
5.3 Vortex dynamics in inhomogeneous potential
In this subsection, we study numerically the vortex dynamics in the CGLE dy-
namics with Dirichlet BC under inhomogeneous potential. Analogous to the one
studied in section 3.4, the external potential and cases studied are chosen as the
same one presented in section 3.4.
Fig. 5.20, shows the trajectory and time evolution of the distance between
the vortex center and potential center for different ε for case I and II, as well as
trajectory of vortex center for different ε of the vortices for case III. From this figure
and additional numerical experiment not shown here for brevity, we can see that:
(i). For the single vortex, it moves monotonically toward the points xp = (x0c , y0c ),
where the external potential V(x) attains its minimum value (cf. Fig. 5.20 (a) &
(b)), which shows clearly the pinning effect. Moreover, the trajectory depend on
the type of the potential V (x). The speed that vortex move to xp as well as the
final location that vortex stay steady depend on the value of ε (cf. Fig. 5.20 (a)
& (b)). The smaller the ε is, the closer the final location to xp and the faster the
5.3 Vortex dynamics in inhomogeneous potential 113
(a) 0 0.2 0.4
0
0.2
0.4
x
y
0 0.5 10
0.3
0.6
t
|xε 1−
O|
ε=1/16
ε=1/25
ε=1/32
ε=1/16
ε=1/25
ε=1/32
(b) 0 0.2 0.4
0
0.2
0.4
x
y
ε=1/16
ε=1/25
ε=1/32
0 0.6 1.20
0.3
0.6
t
|xε 1−
O|
ε=1/16
ε=1/25
ε=1/32
(c) -0.3 0 0.3
-0.3
0
0.3
x
y
ε=1/16
ε=1/32
ε=1/50
0 1.4 2.80
0.17
0.34
t
|xε 1−
O|
ε=1/16ε=1/32ε=1/50
Figure 5.20: Trajectory and time evolution of the distance between the vortex
center different ε for case I-III ((a)-(c)) in section 5.3.
vortex move to it. (ii). As ε → 0, the trajectory seems to converge, and the final
location of the vortex seems to converge to points xp, which are the position that
the potential attain its minimum. (iii). For the vortex pair, well, the phenomena is
quite different from the one in the GLE dynamics. The vortices will always move
close to each other and the point xp firstly, which show the pinning effect. However,
they will move apart from each other and the point xp after some time. And the
smaller the ε is, the farther the vortex pair sit away from each other and the points
xp.
5.4 Conclusion 114
5.4 Conclusion
In this chapter, by applying the efficient and accurate numerical methods pro-
posed in chapter 2 to simulate complex Ginzburg-Landau equation (CGLE) with a
dimensionless parameter 0 < ε < 1 on bounded domains with either Dirichlet or
homogenous Neumann BC and its corresponding reduced dynamical laws (RDLs),
we studied numerically quantized vortex interaction in CGLE with/without impuri-
ties for superconductivity and compared numerically patterns of vortex interaction
between the CGLE dynamics and its corresponding reduced dynamical laws under
different initial setups. We find that vortex dynamics in the CGLE is somehow the
combination of that in GLE and NLSE.
For the CGLE under a homogeneous potential, based on extensive numerical
results, we verified that the dynamics of vortex centers under the CGLE dynamics
converges to that of the reduced dynamical laws when ε → 0 before they collide
and/or move out of the domain. Certainly, after either vortices collide with each
other or move out of the domain, the RDLs are no longer valid; however, the dy-
namics and interaction of quantized vortices are still physically interesting and they
can be obtained from the direct numerical simulations for the CGLE with fixed
ε > 0 even after they collide and/or move out of the domain. We also identified the
parameter regimes where the RDLs agree with qualitatively and/or qualitatively as
well as fail to agree with those from the CGLE dynamics. In the validity regimes,
the RDL is still valid under small perturbation in the initial data due to the dissipa-
tive nature of the CGLE. Some very interesting nonlinear phenomena related to the
quantized vortex interactions in the CGLE for superconductivity were also observed
from our direct numerical simulation results of CGLE. Different steady state pat-
terns of vortex clusters under the CGLE dynamics were obtained numerically. From
our numerical results, we observed that boundary conditions and domain geome-
try affect significantly on vortex dynamics and interaction, which showed different
interaction patterns compared to those in the whole space case [160, 161].
For the CGLE in an inhomogeneous potential under the Dirichlet BC, by directly
5.4 Conclusion 115
simulate the GLE, we find that vortices move in a quite different ways from that
in the homogeneous case. The vortices basically move toward critical points of the
inhomogeneous potential in the limiting process ε → 0, which show the pinning
effect that caused by the impurities given by the inhomogeneities.
Chapter 6
Numerical methods for GPE with angular
momentum
In this chapter, we first review how to reduce the 3D GPE (1.10) with strongly
anisotropic confining potential V (x) into 2D GPE, then we propose a simple, effi-
cient and accurate numerical method for simulating the dynamics of rotating Bose-
Einstein condensates (BECs) in a rotational frame with/without a long-range dipole-
dipole interaction. We then apply the numerical method to test the dynamical laws
of rotating BECs such as the dynamics of condensate width, angular momentum
expectation and center-of-mass, and to investigate numerically the dynamics and
interaction of quantized vortex lattices in rotating BECs without/with the long-
range dipole-dipole interaction.
6.1 GPE with angular momentum
In many physical experiments of rotating BEC, the external trap V (x) in (1.10)
is strongly confined in the z-direction, i.e.,
V (x) = V2(x, y) +z2
2ε4, x ∈ R3, (6.1)
116
6.1 GPE with angular momentum 117
with 0 < ε ≪ 1 a given dimensionless parameter [14], resulting in a pancake-shaped
BEC. Similar to the case of a non-rotating BEC, formally the GPE (1.10) or (1.13)-
(1.14) can effectively be approximated by a two-dimensional (2D) GPE as [17,18,41]:
i∂tψ(x⊥, t) =
[−1
2∇2
⊥ + V2(x⊥) +κ+ λ(3n2
3 − 1)
ε√2π
|ψ|2 − 3λ
2ϕ− ΩLz
]ψ, (6.2)
ϕ = ϕ(x⊥, t) =(∂n⊥n⊥
− n23∇2
⊥)u(x⊥, t), x⊥ = (x, y)T ∈ R2, t ≥ 0, (6.3)
where ∇⊥ = (∂x, ∂y)T , ∇2
⊥ = ∂xx + ∂yy, n⊥ = (n1, n2)T , ∂n⊥
= n⊥ · ∇⊥, ∂n⊥n⊥=
∂n⊥(∂n⊥
) and
u(x⊥, t) = Gε ∗ |ψ|2, Gε(x⊥) =1
(2π)3/2
∫
R
e−s2/2
√|x⊥|2 + ε2s2
ds, x⊥ ∈ R2. (6.4)
The above problem (6.2)-(6.3) with (6.4) is usually called surface adiabatic model
(SAM) for a rotating BEC with dipole-dipole interaction in 2D. Furthermore, taking
ε→ 0+ in (6.4), we obtain
Gε(x⊥) →1
2π|x⊥|:= G0(x⊥), x⊥ ∈ R2. (6.5)
This, together with (6.4), implies that when ε → 0+,
u(x⊥, t) =1
2π|x⊥|∗ |ψ|2 ⇐⇒ (−∇2
⊥)1/2u = |ψ|2 with lim
|x⊥|→∞u(x⊥, t) = 0. (6.6)
The problem (6.2)-(6.3) with (6.6) is usually called surface density model (SDM) for
a rotating BEC with dipole-dipole interaction in 2D. Note that even for the SDM
we retain the ε-dependence in (6.2).
In fact, the GPE (1.10) or (1.13) in 3D and the SAM or SDM in 2D can be
written in a unified way in d-dimensions (d = 2 or 3) with x = (x, y)T when d = 2
and x = (x, y, z)T when d = 3:
i∂tψ(x, t) =
[−1
2∇2 + V (x) + β|ψ|2 + ηϕ(x, t)− ΩLz
]ψ(x, t), (6.7)
ϕ(x, t) = Lnu(x, t), u(x, t) = G ∗ |ψ|2, x ∈ Rd, t ≥ 0, (6.8)
where V (x) = V2(x) when d = 2 and
β =
κ+λ(3n23−1)
ε√2π
,
κ− λ,η =
−3λ/2,
−3λ,Ln =
∂n⊥n⊥− n2
3∇2, d = 2,
∂nn, d = 3,(6.9)
6.2 Dynamical properties 118
G(x) =
1/2π|x|,Gε(x),
1/4π|x|,
⇐⇒ G(ξ) =
1/|ξ|, d = 2 & SDM,
12π2
∫R
e−ε2s2/2
|ξ|2+s2 ds, d = 2 & SAM,
1/|ξ|2, d = 3,
(6.10)
where f(ξ) denotes the Fourier transform of the function f(x) for x, ξ ∈ Rd. For
studying the dynamics of a rotating BEC, the following initial condition is used:
ψ(x, 0) = ψ0(x), x ∈ Rd, with ‖ψ0‖2 :=∫
Rd
|ψ0(x)|2 dx = 1. (6.11)
We remark here that in most BEC experiments, the following dimensionless har-
monic potential is used
V (x) =1
2
γ2xx2 + γ2yy
2, d = 2,
γ2xx2 + γ2yy
2 + γ2zz2, d = 3,
(6.12)
where γx > 0, γy > 0 and γz > 0 are dimensionless constants proportional to the
trapping frequencies in x-, y- and z-direction, respectively.
6.2 Dynamical properties
In this section, we analytically study the dynamics of rotating dipolar BECs. We
present dynamical laws, including the conservation of angular momentum expecta-
tion, the dynamics of condensate widths and the dynamics of the center of mass.
The results are quite similar to the ones in [18, 25].
6.2.1 Conservation of mass and energy
The GPE in (6.7)–(6.11) has two important invariants: the mass (or normaliza-
tion) of the wave function, which is defined as
N(t) := ‖ψ(·, t)‖2 :=∫
Rd
|ψ(x, t)|2dx ≡∫
Rd
|ψ(x, 0)|2dx = 1, t ≥ 0, (6.13)
and the energy per particle
E(t) := E(ψ(·, t)) =∫
Rd
[1
2|∇ψ|2 + V (x)|ψ|2 + β
2|ψ|4 + η
2ϕ|ψ|2 − Ωψ∗Lzψ
]dx
≡ E(ψ(·, 0)) = E(ψ0), t ≥ 0, (6.14)
6.2 Dynamical properties 119
where f ∗ denotes the conjugate of the complex-valued function f . Stationary states,
corresponding to critical points of the energy defined in (6.14), play an important
role in the study of rotating dipolar BECs. Usually, to find stationary states φs(x),
one can use the ansatz
ψ(x, t) = e−iµstφs(x), x ∈ Rd, t ≥ 0, (6.15)
where µs ∈ R is the chemical potential. Substituting (6.15) into (6.7) yields the
nonlinear eigenvalue problem
µsφs(x) =
[−1
2∇2 + V (x) + β|φs|2 + ηϕs − ΩLz
]φs(x), x ∈ Rd, (6.16)
ϕs(x) = Lnus(x), us(x) = G ∗ |φs|2, x ∈ Rd, (6.17)
under the constraint
‖φs‖2 =∫
Rd
|φs(x)|2dx = 1. (6.18)
Thus, by solving the constrained nonlinear eigenvalue problem (6.16)–(6.18), one
can find the stationary states of rotating dipolar BECs. In physics literature, the
stationary state with the lowest energy is called ground state, while those with larger
energy are called excited states.
6.2.2 Conservation of angular momentum expectation
The angular momentum expectation of a condensate is defined as [18, 25]
〈Lz〉(t) =∫
Rd
ψ∗(x, t)Lzψ(x, t) dx, t ≥ 0. (6.19)
This quantity is often used to measure the vortex flux. The following lemma de-
scribes the dynamics of angular momentum expectation in rotating dipolar BECs.
Lemma 6.2.1. Suppose that ψ(x, t) solves the GPE (6.7)–(6.11) with V (x) chosen
as the harmonic potential (6.12). Then we have
d〈Lz〉(t)dt
= (γ2x − γ2y)
∫
Rd
xy|ψ|2dx− η
∫
Rd
|ψ|2 [(x∂y − y∂x)ϕ] dx, t ≥ 0. (6.20)
6.2 Dynamical properties 120
Furthermore, the angular momentum expectation is conserved, i.e.,
〈Lz〉(t) ≡ 〈Lz〉(0), t ≥ 0, (6.21)
if the following two conditions are satisfied: (i). γx = γy; (ii). any one of the
following conditions hold true: (a). η = 0; (b). in 3D, n = (0, 0, 1)T ; (c). in 2D, n =
(0, 0, 1)T and ψ0 satisfies ψ0(x) = f(r)eimθ. That is, in a radially symmetric trap in
2D or a cylindrically symmetric trap in 3D, the angular momentum expectation is
conserved when either there is no dipolar interaction or the dipole axis is parallel to
the z-axis in 3-D or in 2-D with a radially symmetric or central vortex-type initial
data.
Proof. Differentiating (6.19) with respect to t, integrating by parts and taking (6.7)
into account, we get
d〈Lz〉(t)dt
=− i
∫
R3
[ψ∗t (x∂y − y∂x)ψ + ψ∗(x∂y − y∂x)ψt
]dx
=
∫
R3
[(−iψ∗
t )(x∂y − y∂x)ψ + (iψt)(x∂y − y∂x)ψ∗]dx
=
∫
R3
[− 1
2
(∇2ψ∗(x∂y − y∂x)ψ +∇2ψ(x∂y − y∂x)ψ
∗)
+(V (x) + βd|ψ|2 + ηϕ
) [(x∂y − y∂x)|ψ|2
] ]dx
=−∫
R3
|ψ|2[(x∂y − y∂x) (V (x) + ηϕ)
]dx, t ≥ 0. (6.22)
Substituting (6.12) into (6.22) leads to (6.20) immediately. In 3D, due to (1.14), the
6.2 Dynamical properties 121
second term in (6.20) further becomes
− η
∫
R3
|ψ|2(x∂y − y∂x)ϕdx = η
∫
R3
∇2u(x∂y − y∂x) (∂nnu) dx
=− η
∫
R3
(∂ny∇2u+ y∇2∂nu)(∂x∂nu)dx+ η
∫
R3
(∂nx∇2u+ x∇2∂nu)(∂y∂nu) dx
=η
∫
R3
[n1∂y(∂nu)− n2∂x(∂nu)
]∇2u dx− η
∫
R3
[x∂y(∂nu)− y∂x(∂nu)
]∇2(∂nu) dx
=η
∫
R3
[n1∂y(∂nu)− n2∂x(∂nu)
]|ψ|2 dx+ η
∫
R3
[y(∇∂nu)∂x(∇ · ∂nu)
+∇y∇(∂nu)∂x(∂nu)− x(∇∂nu)∂y(∇∂nu)−∇x∇(∂nu)∂y(∂nu)]dx
=η
∫
R3
[n1∂y(∂nu)− n2∂x(∂nu)
]|ψ|2 dx+ η
∫
R3
(∇∂nu)(y∂x − x∂y)(∇∂nu)dx
=η
∫
R3
[n1∂y(∂nu)− n2∂x(∂nu)
]|ψ|2 dx, t ≥ 0. (6.23)
Thus, in a radially symmetric trap, i.e., γx = γy, if there is λ = 0 or n = (0, 0, 1)T ,
we get
d〈Lz〉(t)dt
= 0, t ≥ 0, (6.24)
from (6.20) and (6.23), which implies the conservation of 〈Lz〉 in (6.21).
Look into (6.20), claims in (ii) (a) is clearly and (c) is straightforward due to the
radial symmetry of the solution of ψ(x, t) under the condition given there.
6.2.3 Dynamics of condensate width
The condensate width of a BEC in α-direction (where α = x, y, z or r =√x2 + y2)
is defined by
σα(t) =√δα(t), t ≥ 0, where δα(t) =
∫
Rd
α2|ψ(x, t)|2dx. (6.25)
In particular, when d = 2, we have the following lemma for its dynamics [18]:
Lemma 6.2.2. Consider two-dimensional BECs with radially symmetric harmonic
trap (6.12), i.e., d = 2 and γx = γy := γr. If η = 0, then for any initial datum
ψ0(x) in (6.11), it holds for t ≥ 0
δr(t) =E(ψ0) + Ω〈Lz〉(0)
γ2r[1− cos(2γrt)] + δ(0)r cos(2γrt) +
δ(1)r
2γrsin(2γrt), (6.26)
6.2 Dynamical properties 122
where δr(t) := δx(t) + δy(t), δ(0)r := δx(0) + δy(0) and δ
(1)r := δx(0) + δy(0). Further-
more, if the initial condition ψ0(x) is radially symmetric, we have for t ≥ 0
δx(t) = δy(t) =1
2δr(t)
=E(ψ0) + Ω〈Lz〉(0)
2γ2x[1− cos(2γxt)] + δ(0)x cos(2γxt) +
δ(1)x
2γxsin(2γxt). (6.27)
Thus, in this case the condensate widths σx(t) and σy(t) are periodic functions with
frequency doubling trapping frequency.
Proof. Differentiating (6.25) with respect to t, integrating by parts and taking (6.7)
into account, we obtain
dδα(t)
dt=
∫
Rd
α2 (ψ∗tψ + ψ∗ψt) dx = i
∫
Rd
α2[(−iψ∗
t )ψ − ψ∗(iψt)]dx
= i
∫
Rd
[−α
2
2
(ψ∇2ψ∗ − ψ∗∇2ψ
)− iΩα2 (x∂y − y∂x) |ψ|2
]dx
=
∫
Rd
[iα(ψ∂αψ
∗ − ψ∗∂αψ)− 2Ωα|ψ|2(x∂y − y∂x)α]dx, t ≥ 0. (6.28)
Similarly, differentiating (6.28) with respect to t, integrating by parts and noticing
(6.7), we have
d2δα(t)
dt2=
∫
Rd
α
[i(ψtψ
∗α + ψψ∗
αt − ψ∗tψα − ψ∗ψαt
)− 2Ω(ψ∗
tψ + ψ∗ψt)(x∂y − y∂x)α
]dx
=
∫
Rd
[2α((iψt)ψ
∗α + (−iψ∗
t )ψα
)+(ψ∗(iψt) + ψ(−iψ∗
t ))
− 2iΩα(x∂yα− y∂xα)((−iψ∗
t )ψ − ψ∗(iψt))]dx
=
∫
Rd
[α(− (ψ∗
α∇2ψ + ψα∇2ψ∗) + 2(V (x) + β|ψ|2 + ηϕ)∂α|ψ|2
+ 2iΩ(ψ∗α(x∂y − y∂x)ψ − ψα(x∂y − y∂x)ψ
∗))+(− 1
2(ψ∗∇2ψ + ψ∇2ψ∗)
+ 2(V (x) + β|ψ|2 + ηϕ)|ψ|2 − iΩ(ψ(x∂y − y∂x)ψ∗ − ψ∗(x∂y − y∂x)ψ)
)
+ 2iΩα(x∂yα− y∂xα)(12(ψ∇2ψ∗ − ψ∗∇2ψ) + iΩ(x∂y − y∂x)|ψ|2
)]dx;
6.2 Dynamical properties 123
Moreover, we obtain for t ≥ 0
d2δα(t)
dt2=
∫
Rd
[(2|ψα|2 − |∇ψ|2 − 2(V (x) + β|ψ|2 + ηϕ)|ψ|2 + β|ψ|4
− 2α|ψ|2∂α(V (x) + ηϕ) + 2iΩ(∂yα− ∂xα)ψ∗(x∂y + y∂x)ψ
− 2iΩψ∗(x∂y − y∂x)ψ)+(|∇ψ|2 + 2(V (x) + β|ψ|2 + ηϕ)|ψ|2
+ 2iΩψ∗(x∂y − y∂x)ψ)+ (∂yα− ∂xα)
(2iΩψ∗(x∂y + y∂x)ψ
+ 2Ω2(x2 − y2)|ψ|2)]dx
= −2γ2αδα(t) +
∫
R2
[2|ψα|2 + β|ψ|4 − 2ηα|ψ|2∂αϕ
+ (∂yα− ∂xα)(4iΩ(x∂y + y∂x)ψ + 2Ω2(x2 − y2)|ψ|2
)]dx. (6.29)
Hence when d = 2, if γx = γy = γr and η = 0, combine with (6.29), (6.14) and
lemma 6.2.1, we have
d2δr(t)
dt2=d2δx(t)
dt2+d2δy(t)
dt2= −4γ2r δr(t) + 2
∫
R2
[|∇ψ|2 + β|ψ|4
]dx
=− 4γ2rδr(t) + 4
∫
R2
[1
2|∇ψ|2 + V (x)|ψ|2 + β
2|ψ|4 − ΩRe(ψ∗Lzψ)
]dx
+ 4Ω
∫
R2
Re(ψ∗Lzψ)dx
=− 4γ2rδr(t) + 4E(ψ0) + 4Ω〈Lz〉(0), t ≥ 0. (6.30)
Thus, (6.26) is the unique solution of the second order ODE (6.30) with the
initial condition δr(0) = δ(0)r and δr(0) = δ
(1)r .
Furthermore, if ψ0 has radial symmetric structure, the solution ψ(x, t) is also
radial symmetric since γx = γy, and can be rewritten in the form of
ψ(x, t) = f(r, t)eimθ(t)
and thus
δx(t) =
∫
Rd
x2|φ|2dx =
∫ ∞
0
∫ 2π
0
r2 cos2 θ|f(r, t)|2rdθdr
= π
∫ ∞
0
r2|f(r, t)|2rdr =∫ ∞
0
∫ 2π
0
r2 sin2 θ|f(r, t)|2rdθdr
=
∫
Rd
y2|φ|2dx = δy(t) =1
2δr(t). (6.31)
6.2 Dynamical properties 124
6.2.4 Dynamics of center of mass
We define the center of mass of a condensate at any time t by
xc(t) =
∫
Rd
x |ψ(x, t)|2dx, t ≥ 0. (6.32)
The following lemma describes the dynamics of the center of mass.
Lemma 6.2.3. Suppose that ψ(x, t) solves the GPE (6.7)–(6.11) with V (x) chosen
as the harmonic potential (6.12). Then for any given initial data ψ0, the dynamics
of the center of mass are governed by the following second-order ODEs:
xc(t)− 2ΩJ xc(t) + (Λ + Ω2J2)xc(t) = 0, t ≥ 0, (6.33)
xc(0) = x(0)c :=
∫
Rd
x|ψ0|2dx, (6.34)
xc(0) = x(1)c :=
∫
Rd
Im (ψ∗0∇ψ0) dx− ΩJx(0)
c , (6.35)
where Im(f) denotes the imaginary part of the function f and the matrices
J =
0 1
−1 0
, Λ =
γ2x 0
0 γ2y
, for d = 2,
or
J =
0 1 0
−1 0 0
0 0 0
, Λ =
γ2x 0 0
0 γ2y 0
0 0 γ2z
, for d = 3.
Proof. For simplicity, we consider d = 3 in the following proof. Differentiating (6.32)
with respect to t, integrating by parts and taking (6.7) into account, we obtain
dxc(t)
dt=
∫
R3
x(ψ∗t ψ + ψ∗ψt)dx = i
∫
R3
x[(−iψ∗
t )ψ − ψ∗(iψt)]dx
= i
∫
R3
[− 1
2x(ψ∇2ψ∗ − ψ∗∇2ψ
)− iΩx(x∂y − y∂x)|ψ|2
]dx
=
∫
R3
[− iψ∗∇ψ − Ω|ψ|2(x∂y − y∂x)x
]dx
=
∫
R3
(− iψ∗∇ψ +ΩJx|ψ|2
)dx = ΩJxc(t)− i
∫
R3
(ψ∗∇ψ) dx, t ≥ 0. (6.36)
6.2 Dynamical properties 125
Differentiating (6.36) with respect to t and following the similar procedures, we have
d2xc(t)
dt2− ΩJ
dxc(t)
dt
= −i∫
R3
(ψ∗t∇ψ + ψ∗∇ψt) dx =
∫
R3
[(−iψ∗
t )∇ψ + (iψt)∇ψ∗]dx
=
∫
R3
[− 1
2
(∇2ψ∗∇ψ +∇2ψ∇ψ∗)+ (V (x) + ηϕ)∇|ψ|2
− iΩ((x∂y − y∂x)ψ
∗∇ψ − (x∂y − y∂x)ψ∇ψ∗)]dx
= −∫
R3
|ψ|2∇ [V (x) + ηϕ] dx− iΩJ
∫
R3
(ψ∗∇ψ) dx. (6.37)
Noticing (6.36) and V (x) defined in (6.12), we can rewrite the ODE (6.37) as
d2xc(t)
dt2− ΩJ
dxc(t)
dt
= −Λ
∫
R3
x|ψ|2dx− ΩJ
(dxc(t)
dt+ ΩJxc(t)
)+ η
∫
R3
(∇2u)∇∂nnu dx
= −Λ
∫
R3
x|ψ|2dx− ΩJ
(dxc(t)
dt+ ΩJxc(t)
)− η
∫
R3
∇|∇(∂nu)|2 dx
= −Λxc(t) + ΩJdxc(t)
dt+ Ω2xc(t), t ≥ 0. (6.38)
Combining (6.38) with (6.32) and (6.36) at time t = 0, we can get the ODEs (6.33)–
(6.35).
Lemma 6.2.3 shows that the dynamics of the center of mass depends on the trapping
frequencies and the angular velocity, but it is independent of the interaction strength
constants β and η in (6.7). For analytical solutions to the second-order ODEs (6.33)-
(6.35), we refer to [159].
6.2.5 An analytical solution under special initial data
From Lemma 6.2.3, we can construct an analytical solution to the GPE (6.7)–
(6.11) when the initial data is chosen as a stationary state with its center shifted.
Lemma 6.2.4. Suppose V (x) in (6.7) is chosen as the harmonic potential (6.12)
and the initial condition ψ0(x) in (6.11) is chosen as
ψ0(x) = φs(x− x0), x ∈ Rd, (6.39)
6.3 GPE under a rotating Lagrangian coordinate 126
where x0 ∈ Rd is a given point and φs(x) is a stationary state defined in (6.16)–
(6.18) with chemical potential µs, then the exact solution of (6.7)–(6.11) can be
constructed as
ψ(x, t) = φs(x− xc(t)) e−iµst eiw(x,t), x ∈ Rd, t ≥ 0, (6.40)
where xc(t) satisfies the ODE (6.33) with
xc(0) = x0, xc(0) = −ΩJx0, (6.41)
and w(x, t) is linear in x, i.e.,
w(x, t) = c(t) · x + g(t), c(t) = (c1(t), . . . , cd(t))T , x ∈ Rd, t ≥ 0
for some functions c(t) and g(t). Thus, up to phase shifts, ψ remains a stationary
state with shifted center at all times.
6.3 GPE under a rotating Lagrangian coordinate
In this section, we first introduce a coordinate transformation and derive the
GPE in transformed coordinates. Then we reformulate the dynamical quantities
studied in Section 6.2 in the new coordinate system.
6.3.1 A rotating Lagrangian coordinate transformation
For any time t ≥ 0, let A(t) be an orthogonal rotational matrix defined as
A(t) =
cos(Ωt) sin(Ωt)
− sin(Ωt) cos(Ωt)
, if d = 2, (6.42)
and
A(t) =
cos(Ωt) sin(Ωt) 0
− sin(Ωt) cos(Ωt) 0
0 0 1
, if d = 3. (6.43)
6.3 GPE under a rotating Lagrangian coordinate 127
It is easy to verify that A−1(t) = AT (t) for any t ≥ 0 and A(0) = I with I the
identity matrix. For any t ≥ 0, we introduce the rotating Lagrangian coordinates x
as [11, 65, 73]
x = A−1(t)x = AT (t)x ⇔ x = A(t)x, x ∈ Rd, (6.44)
and denote the wave function in the new coordinates as φ := φ(x, t)
φ(x, t) := ψ(x, t) = ψ (A(t)x, t) , x ∈ Rd, t ≥ 0. (6.45)
In fact, here we refer the Cartesian coordinates x as the Eulerian coordinates and
Fig. 6.1 depicts the geometrical relation between the Eulerian coordinates x and
the rotating Lagrangian coordinates x for any fixed t ≥ 0.
Ωt
y
x
y
x
Figure 6.1: Cartesian (or Eulerian) coordinates (x, y) (solid) and rotating La-
grangian coordinates (x, y) (dashed) in 2D for any fixed t ≥ 0.
Using the chain rule, we obtain the derivatives:
∂tφ(x, t) = ∂tψ(x, t) +∇ψ(x, t) ·(A(t)x
)= ∂tψ(x, t)− Ω(x∂y − y∂x)ψ(x, t),
∇φ(x, t) = A−1(t)∇ψ(x, t), ∇2φ(x, t) = ∇2ψ(x, t).
Substituting them into (6.7)–(6.11) gives the following d-dimensional GPE in the
rotating Lagrangian coordinates:
i∂tφ(x, t) =
[−1
2∇2 +W (x, t) + β|φ|2 + ηϕ(x, t)
]φ(x, t), x ∈ Rd, t > 0, (6.46)
ϕ(x, t) = Lm(t)u(x, t), u(x, t) = G ∗ |φ|2, x ∈ Rd, t ≥ 0, (6.47)
6.3 GPE under a rotating Lagrangian coordinate 128
where G is defined in (6.10) and
W (x, t) = V (A(t)x), x ∈ Rd, (6.48)
Lm(t) =
∂m⊥(t)m⊥(t) − n23∇2, d = 2,
∂m(t)m(t), d = 3,
t ≥ 0, (6.49)
with m(t) ∈ R3 and m⊥(t) ∈ R2 defined as
m(t) =
m1(t)
m2(t)
m3(t)
:= A−1(t)n =
n1 cos(Ωt)− n2 sin(Ωt)
n1 sin(Ωt) + n2 cos(Ωt)
n3
, t ≥ 0, (6.50)
and m⊥(t) = (m1(t), m2(t))T , respectively. The initial data transforms as
φ(x, 0) = ψ(x, 0) = ψ0(x) := φ0(x) = φ0(x), x = x ∈ Rd. (6.51)
We remark here again that if V (x) in (6.7) is a harmonic potential as defined in
(6.12), then the potential W (x, t) in (6.46) has the form
W (x, t) =1
4
ω1(x2 + y2) + ω2 [(x
2 − y2) cos(2Ωt) + 2xy sin(2Ωt)] , d = 2,
ω1(x2 + y2) + ω2 [(x
2 − y2) cos(2Ωt) + 2xy sin(2Ωt)] + 2γ2z z2, d = 3,
where ω1 = γ2x + γ2y and ω2 = γ2x − γ2y . It is easy to see that when γx = γy := γr,
i.e., radially and cylindrically symmetric harmonic trap in 2D and 3D, respectively,
we have ω1 = 2γ2r and ω2 = 0 and thus the potential W (x, t) = V (x) becomes
time-independent.
In contrast to (6.7), the GPE (6.46) does not have an angular momentum rota-
tion term, which enables us to develop simple and efficient numerical methods for
simulating the dynamics of rotating dipolar BEC in Section 6.4.
6.3.2 Dynamical quantities
In the above, we introduced rotating Lagrangian coordinates and cast the GPE
in the new coordinate system. Next we consider the dynamical laws in terms of the
new wave function φ(x, t).
6.3 GPE under a rotating Lagrangian coordinate 129
Mass and energy. In rotating Lagrangian coordinates, the conservation of mass
(6.13) yields
‖ψ(·, t)‖2 :=∫
Rd
|ψ(x, t)|2dx =
∫
Rd
|φ(x, t)|2dx = ‖φ(·, t)‖2 ≡ 1, t ≥ 0. (6.52)
The energy defined in (6.14) becomes
E(φ(·, t)) =∫
Rd
[1
2|∇φ|2 +W (x, t)|φ|2 + β
2|φ|4 + η
2ϕ|φ|2
]dx
−∫
Rd
∫ t
0
[∂sW (x, s) +
η
2(∂sLm(s))u(x, s)
]|φ|2ds dx
≡ E(φ(·, 0)), t ≥ 0, (6.53)
where u is given in (6.47). Specifically, it holds
∂tLm(t) = 2
∂AT (t)n⊥
∂AT (t)n⊥, d = 2,
∂AT (t)n∂AT (t)n, d = 3.
Angular momentum expectation. The angular momentum expectation in the
new coordinates becomes
〈Lz〉(t) = −i∫
Rd
ψ∗(x, t)(x∂y − y∂x)ψ(x, t) dx
= −i∫
Rd
φ∗(x, t)(x∂y − y∂x)φ(x, t) dx, t ≥ 0, (6.54)
which has the same form as (6.19) in the new coordinates of x ∈ Rd and the
wave function φ(x, t). Indeed, if we denote Lz as the z-component of the angular
momentum in the rotating Lagrangian coordinates, we have Lz = −i(x∂y − y∂x) =
−i(x∂y − y∂x) = Lz, i.e., the coordinate transform does not change the angular
momentum in z-direction. In addition, noticing that for any t ≥ 0 it holds φ(x, t) =
ψ(x, t) and |A(t)| = 1 for any t ≥ 0 immediately yields (6.54).
Condensate width. After the coordinate transform, it holds
δr(t) =
∫
Rd
(x2 + y2)|ψ|2dx =
∫
Rd
(x2 + y2)|φ|2dx = δx(t) + δy(t), (6.55)
δz(t) =
∫
Rd
z2|ψ|2dx =
∫
Rd
z2|φ|2dx = δz(t), (6.56)
6.3 GPE under a rotating Lagrangian coordinate 130
for any t ≥ 0.
Center of mass. The center of mass in rotating Lagrangian coordinates is defined
as
xc(t) =
∫
Rd
x |φ(x, t)|2dx, t ≥ 0. (6.57)
Since det(A(t)) = 1 for any t ≥ 0, it holds that xc(t) = A(t)xc(t) for any time t ≥ 0.
In rotating Lagrangian coordinates, we have the following analogue of Lemma 6.2.4:
Lemma 6.3.1. Suppose V (x) in (6.7) is chosen as the harmonic potential (6.12)
and the initial condition φ0(x) in (6.51) is chosen as
φ0(x) = φs(x− x0), x ∈ Rd, (6.58)
where x0 is a given point in Rd and φs(x) is a stationary state defined in (6.16)–
(6.18) with chemical potential µs, then the exact solution of (6.46)–(6.47) is of the
form
φ(x, t) = φs(x− xc(t)) e−iµst eiw(x,t), t > 0, (6.59)
where xc(t) satisfies the second-order ODEs:
¨xc(t) + AT (t)ΛA(t) xc(t) = 0, t ≥ 0, (6.60)
xc(0) = x0, ˙xc(0) = 0, (6.61)
with the matrix Λ defined in Lemma 6.2.3 and w(x, t) is linear in x, i.e.,
w(x, t) = c(t) · x + g(t), c(t) = (c1(t), . . . , cd(t))T , x ∈ Rd, t ≥ 0.
We have seen that the form of the transformation matrix A(t) in (6.43) is such
that the coordinate transformation does not affect the quantities in z-direction, e.g.
〈Lz〉(t), σz(t) and zc(t).
6.4 Numerical methods 131
6.4 Numerical methods
To study the dynamics of rotating dipolar BECs, in this section we propose
a simple and efficient numerical method for discretizing the GPE (6.46)–(6.51) in
rotating Lagrangian coordinates. The detailed discretizations for both the 2D and
3D GPEs are presented. Here we assume Ω 6= 0, and for Ω = 0, we refer to
[15, 17, 41, 151].
In practical computations, we first truncate the whole space problem (6.46)–
(6.51) to a bounded computational domain D ⊂ Rd and consider
i∂tφ(x, t) = −1
2∇2φ+W (x, t)φ+ β|φ|2φ+ ηϕφ, x ∈ D, t > 0, (6.62)
ϕ(x, t) = Lm(t)u(x, t), u(x, t) =
∫
Rd
G(x− y)ρ(y, t) dy, x ∈ D, t > 0; (6.63)
where
ρ(y, t) =
|φ(y, t)|2, y ∈ D,0, otherwise,
y ∈ Rd.
The initial condition is given by
φ(x, 0) = φ0(x), x ∈ D. (6.64)
The boundary condition to (6.62) will be chosen based on the kernel function G
defined in (6.10). Due to the convolution in (6.63), it is natural to consider using
the Fourier transform to compute u(x, t). However, from (6.10) and (6.52), we
know that limξ→0 G(ξ) = ∞ and |φ|2(ξ = 0) 6= 0. As noted for simulating dipolar
BECs in 3D [17, 37, 131], there is a numerical locking phenomena, i.e., numerical
errors will be bounded below no matter how small the mesh size is, when one uses
the Fourier transform to evaluate u(x, t) and/or ϕ(x, t) numerically in (6.63). As
noticed in [14,17], the second (integral) equation in (6.63) can be reformulated into
the Poisson equation (1.14) and square-root-Poisson equation (6.6) for 3D and 2D
SDM model, respectively. With these PDE formulations for u(x, t), we can truncate
them on the domain D and solve them numerically via spectral method with sine
6.4 Numerical methods 132
basis functions instead of Fourier basis functions and thus we can avoid using the
0-modes [17]. Thus in 3D and 2D SDM model, we choose the homogeneous Dirichlet
boundary conditions to (6.62). Of course, for the 2D SAM model, one has to use the
Fourier transform to compute u(x, t), thus we take the periodic boundary conditions
to (6.62).
The computational domain D ⊂ Rd is chosen as D = [a, b] × [c, d] if d = 2 and
D = [a, b]× [c, d]× [e, f ] if d = 3. Due to the confinement of the external potential,
the wave function decays exponentially fast as |x| → ∞. Thus if we choose D to be
sufficiently large, the error from the domain truncation can be neglected. As long as
we solve φ(x, t) in the bounded computational domain D, we obtain a corresponding
solution ψ(x, t) in the domain A(t)D. As shown in Fig. 6.2 for the example of a 2D
domain, although the domains A(t)D for t ≥ 0, are in general different for different
time t, they share a common disk which is bounded by the inner green solid circle in
Fig. 6.2. Thus, the value of ψ(x, t) inside the vertical maximal square (the magenta
area) which lies fully within the inner disk can be calculated easily by interpolation.
6.4.1 Time-splitting method
Next, let us introduce a time-splitting method to discretize (6.62)–(6.64). We
choose a time-step size τ > 0 and define the time sequence as tn = nτ for n ∈ N.
Then from t = tn to t = tn+1, we numerically solve the GPE (6.62) in two steps.
First we solve
i∂tφ(x, t) = −1
2∇2φ(x, t), x ∈ D, tn ≤ t ≤ tn+1 (6.65)
for a time step of length τ , and then we solve
i∂tφ(x, t) =[W (x, t) + β|φ|2 + ηϕ
]φ(x, t), x ∈ D, tn ≤ t ≤ tn+1, (6.66)
ϕ(x, t) = Lm(t)u(x, t), u(x, t) =
∫
Rd
G(x− y)ρ(y, t) dy, (6.67)
for the same time step.
6.4 Numerical methods 133
y
x
D
x
y
t = π
4t = π
2
A(t)D
t = 3π4t = 0
Figure 6.2: The bounded computational domain D in rotating Lagrangian coor-
dinates x (left) and the corresponding domain A(t)D in Cartesian (or Eulerian)
coordinates x (right) when Ω = 0.5 at different times: t = 0 (black solid), t = π4
(cyan dashed), t = π2(red dotted) and t = 3π
4(blue dash-dotted). The two green
solid circles determine two disks which are the union (inner circle) and the intersec-
tion of all domains A(t)D for t ≥ 0, respectively. The magenta area is the vertical
maximal square inside the inner circle.
Equation (6.65) can be discretized in space by sine or Fourier pseudospectral
methods and then integrated exactly in time. If homogeneous Dirichlet boundary
conditions are used, then we choose the sine pseudospectral method to discretize it;
otherwise, the Fourier pseudospectral method is used if the boundary conditions are
periodic. For more details, see e.g. [17, 25].
On the other hand, we notice that on each time interval [tn, tn+1], the problem
(6.66)–(6.67) leaves |φ(x, t)| and hence u(x, t) invariant, i.e., |φ(x, t)| = |φ(x, tn)|and u(x, t) = u(x, tn) for all times tn ≤ t ≤ tn+1. Thus, for t ∈ [tn, tn+1], Eq. (6.66)
reduces to
i∂tφ(x, t) =[W (x, t) + β|φ(x, tn)|2 + η
(Lm(t)u(x, tn)
)]φ(x, t), x ∈ D. (6.68)
Integrating (6.68) in time gives the solution
φ(x, t) = φ(x, tn) exp
[−i(β|φ(x, tn)|2(t− tn) + ηΦ(x, t) +
∫ t
tn
W (x, s)ds
)]
(6.69)
6.4 Numerical methods 134
for x ∈ D and t ∈ [tn, tn+1], where the function Φ(x, t) is defined by
Φ(x, t) =
∫ t
tn
[Lm(s)u(x, tn)
]ds =
(∫ t
tn
Lm(s) ds
)u(x, tn). (6.70)
Plugging (6.50) and (6.49) into (6.70), we get
Φ(x, t) = Ld(t)u(x, tn), x ∈ D, tn ≤ t ≤ tn+1, (6.71)
where
Ld(t) =
[l11e (t)− l33e (t)]∂xx + [l22e (t)− l33e (t)]∂yy + l12e (t)∂xy, d = 2,
l11e (t)∂xx + l22e (t)∂yy + l33e (t)∂zz + l12e (t)∂xy + l13e (t)∂xz + l23e (t)∂yz, d = 3,
with
l11e (t) =
∫ t
tn
m21(s)ds =
∫ t
tn
[n21 cos
2(Ωs) + n22 sin
2(Ωs)− n1n2 sin(2Ωs)]ds
=n21 + n2
2
2(t− tn) +
n21 − n2
2
4Ω[sin(2Ωt)− sin(2Ωtn)] +
n1n2
2Ω[cos(2Ωt)− cos(2Ωtn)] ,
l22e (t) =
∫ t
tn
m22(s)ds =
∫ t
tn
[n22 cos
2(Ωs) + n21 sin
2(Ωs) + n1n2 sin(2Ωs)]ds
=n21 + n2
2
2(t− tn)−
n21 − n2
2
4Ω[sin(2Ωt)− sin(2Ωtn)]−
n1n2
2Ω[cos(2Ωt)− cos(2Ωtn)] ,
l12e (t) = 2
∫ t
tn
m1(s)m2(s)ds =
∫ t
tn
[(n2
1 − n22) sin(2Ωs) + 2n1n2 cos(2Ωt)
]ds
=n21 − n2
2
2Ω[cos(2Ωtn)− cos(2Ωt)] +
n1n2
Ω[sin(2Ωt)− sin(2Ωtn)] ,
l13e (t) = 2n3
∫ t
tn
m1(s)ds = 2n3
∫ t
tn
[n1 cos(Ωs)− n2 sin(Ωs)] ds
=2n3
Ω[n1 [sin(Ωt)− sin(Ωtn)] + n2 [cos(Ωt)− cos(Ωtn)]] ,
l23e (t) = 2n3
∫ t
tn
m2(s)ds = 2n3
∫ t
tn
[n1 sin(Ωs) + n2 cos(Ωs)] ds
=2n3
Ω[n2 [sin(Ωt)− sin(Ωtn)]− n1 [cos(Ωt)− cos(Ωtn)]] ,
l33e (t) =
∫ t
tn
n23 ds = n2
3(t− tn), tn ≤ t ≤ tn+1.
In Section 6.4.2, we will discuss in detail the approximations to Φ(x, t) in (6.71).
In addition, we remark here again that if V (x) in (6.7) is a harmonic potential as
6.4 Numerical methods 135
defined in (6.12), then the definite integral in (6.69) can be calculated analytically
as
∫ t
tn
W (x, s)ds =1
4ω1(x
2 + y2)(t− tn) +H(x, t) +1
2
0, d = 2,
γ2z z2(t− tn), d = 3,
where
H(x, t) =1
4
∫ t
tn
ω2
[(x2 − y2) cos(2Ωs) + 2xy sin(2Ωs)
]ds
=ω2
8Ω
[(x2 − y2
)[sin(2Ωt)− sin(2Ωtn)]− 2xy [cos(2Ωt)− cos(2Ωtn)]
].
Of course, for general external potential V (x) in (6.7), the integral of W (x, s)
in (6.69) might not be found analytically. In this situation, we can simply adopt a
numerical quadrature to approximate it, e.g. the Simpson’s rule can be used as
∫ t
tn
W (x, s) ds ≈ t− tn6
[W (x, tn) + 4W (x,
tn + t
2) +W (x, t)
].
We remark here that, in practice, we always use the second-order Strang splitting
method [143] to combine the two steps in (6.65) and (6.66)–(6.67). For a more
general discussion of the splitting method, we refer the reader to [15, 25, 67].
6.4.2 Computation of Φ(x, t)
In this section, we present approximations to the function Φ(x, t) in (6.71).
From the discussion in the previous subsection, we need only show how to discretize
u(x, tn) in (6.63) and its second-order derivatives in (6.71).
Surface adiabatic model in 2D
In this case, the function u(x, tn) in (6.70) is given by
u(x, tn) =
∫
R2
G(x− y)ρ(y, t) dy, x ∈ D, (6.72)
with the kernel function G defined in the second line of (6.10). To approximate it,
we consider a 2D box D with periodic boundary conditions.
6.4 Numerical methods 136
Let M and K be two even positive integers. Then we make the (approximate)
ansatz
u(x, tn) =
M/2−1∑
p=−M/2
K/2−1∑
q=−K/2ufpq(tn) e
iν1p(x−a)eiν2q (y−c), x = (x, y) ∈ D, (6.73)
where ufpq(tn) is the Fourier coefficient of u(x, tn) corresponding to the frequencies
(p, q) and
ν1p =2pπ
b− a, ν2q =
2qπ
d− c, (p, q) ∈ SMK .
The index set SMK is defined as
SMK =
(p, q) | − M
2≤ p ≤ M
2− 1, −K
2≤ q ≤ K
2− 1
.
We approximate the convolution in (6.72) by a discrete convolution and take its
discrete Fourier transform to obtain
ufpq(tn) = G(ν1p , ν2q ) · (|φn|2)fpq, (p, q) ∈ SMK , (6.74)
where (|φn|2)fpq is the Fourier coefficient corresponding to the frequencies (p, q) of
the function |φ(x, tn)|2, and G(ν1p , ν2q ) are given by (see details in (6.10))
G(ν1p , ν2q ) =
1
2π2
∫ ∞
−∞
e−ε2s2/2
(ν1p)2 + (ν2q )
2 + s2ds, (p, q) ∈ SMK . (6.75)
Since the integrand in (6.75) decays exponentially fast, in practice we can first trun-
cate it to an interval [s1, s2] with |s1|, s2 > 0 sufficiently large and then evaluate the
truncated integral by using quadrature rules, e.g. composite Simpson’s or trape-
zoidal quadrature rule.
Combining (6.71), (6.73) and (6.74), we obtain an approximation of Φ(x, t) in
the solution (6.69) via the Fourier spectral method as
Φ(x, t) =
M/2−1∑
p=−M/2
K/2−1∑
q=−K/2
[L(ν1p , ν
2q , t)G(ν
1p , ν
2q ) · (|φn|2)fpq
]eiν
1p(x−a)eiν
2q (y−c), (6.76)
for time tn ≤ t ≤ tn+1, where the function L(ξ1, ξ2, t) is defined as
L(ξ1, ξ2, t) = −[(l11e (t)− l33e (t)
)ξ21 +
(l22e (t)− l33e (t)
)ξ22 + l12e (t)ξ1ξ2
].
6.4 Numerical methods 137
Surface density model in 2D
In this case, the function u(x, tn) in (6.70) also satisfies the square-root-Poisson
equation in (6.6) which can be truncated on the computational domain D with
homogeneous Dirichlet boundary conditions as
(−∇2)1/2u(x, tn) = |φ(x, tn)|2, x ∈ D; u(x, tn)|∂D = 0. (6.77)
The above problem can be discretized by using a sine pseudospectral method in
which the 0-modes are avoided. Letting M,K ∈ N, we denote the index set
TMK = (p, q) | 1 ≤ p ≤M − 1, 1 ≤ q ≤ K − 1 ,
and define the functions
Up,q(x) = sin(µ1p(x− a)) sin(µ2
q(y − c)), (p, q) ∈ TMK , x = (x, y) ∈ D,
where
µ1p =
pπ
b− a, µ2
q =qπ
d− c, (p, q) ∈ TMK . (6.78)
Assume that
u(x, tn) =
M−1∑
p=1
K−1∑
q=1
uspq(tn)Up,q(x), x ∈ D, (6.79)
where uspq(tn) is the sine transform of u(x, tn) at frequencies (p, q). Substituting
(6.79) into (6.77) and taking sine transform on both sides, we obtain
uspq(tn) =(|φn|2)spq√(µ1
p)2 + (µ2
q)2, (p, q) ∈ TMK , (6.80)
where (|φn|2)spq is the sine transform of |φ(x, tn)|2 at frequencies (p, q).
Combining (6.79), (6.80) and (6.71), we obtain an approximation of Φ(x, t) in
the solution (6.69) via sine spectral method as
Φ(x, t) =M−1∑
p=1
K−1∑
q=1
(|φn|2)spq√(µ1
p)2 + (µ2
q)2
[L(µ1
p, µ2q, t)Up,q(x) + l12e (t)Vp,q(x)
], (6.81)
where the functions L(ξ1, ξ2, t) and Vp,q(x) are defined as
L(ξ1, ξ2, t) = −[(l11e (t)− l33e (t)
)ξ21 +
(l22e (t)− l33e (t)
)ξ22], (6.82)
Vp,q(x) = ∂xyUp,q(x) = µ1pµ
2q cos(µ
1p(x− a)) cos(µ2
q(y − c)), (p, q) ∈ TMK . (6.83)
6.4 Numerical methods 138
Approximations in 3D
In 3D case, again the function u(x, tn) in (6.70) also satisfies the Poisson equation
in (1.14) which can be truncated on the computational domain D with homogeneous
Dirichlet boundary conditions as
−∇2u(x, tn) = |φ(x, tn)|2, x ∈ D; u(x, tn)|∂D = 0. (6.84)
The above problem can be discretized by using a sine pseudospectral method in
which the 0-modes are avoided. Denote the index set
TMKL = (p, q, r) | 1 ≤ p ≤M − 1, 1 ≤ q ≤ K − 1, 1 ≤ r ≤ L− 1
where M,K,L > 0 are integers and define the functions
Up,q,r(x) = sin(µ1p(x− a)) sin(µ2
q(y − c)) sin(µ3r(z − e)), (p, q, r) ∈ TMKL,
where
µ3r = rπ/(f − e), 1 ≤ r ≤ L− 1.
Again, we take the (approximate) ansatz
u(x, tn) =M−1∑
p=1
K−1∑
q=1
L−1∑
r=1
uspqr(tn) Up,q,r(x), x = (x, y, z) ∈ D, (6.85)
where uspqr(tn) is the sine transform of u(x, tn) corresponding to frequencies (p, q, r).
Substituting (6.85) into the Poisson equation (6.84) and noticing the orthogonality
of the sine functions, we obtain
uspqr(tn) =(|φn|2)spqr
(µ1p)
2 + (µ2q)
2 + (µ3r)
2, (p, q, r) ∈ TMKL, (6.86)
where (|φn|2)spqr is the sine transform of |φ(x, tn)|2 corresponding to frequencies
(p, q, r).
Combining (6.71), (6.85) and (6.86), we obtain an approximation of Φ(x, t) in
the solution (6.69) via sine spectral method as
Φ(x, t) =
M−1∑
p=1
K−1∑
q=1
L−1∑
r=1
uspqr(tn)
[L(µ1
p, µ2q, µ
3r, t)Up,q,r(x) + l12e V
(1)p,q,r(x)
+ l13e V(2)p,q,r(x) + l23e V
(3)p,q,r(x)
], x ∈ D, (6.87)
6.5 Numerical results 139
where the functions L(ξ1, ξ2, ξ3, t), V(1)p,q,r(x), V
(2)p,q,r(x) and V
(3)p,q,r(x) (for (p, q, r) ∈
TMKL) are defined as
L(ξ1, ξ2, ξ3, t) = −[l11e (t)ξ21 + l22e (t)ξ22 + l33e (t)ξ23
],
V (1)p,q,r(x) = ∂xyUp,q,r(x) = µ1
pµ2q cos(µ
1p(x− a)) cos(µ2
q(y − c)) sin(µ3r(z − e)),
V (2)p,q,r(x) = ∂xzUp,q,r(x) = µ1
pµ3r cos(µ
1p(x− a)) sin(µ2
q(y − c)) cos(µ3r(z − e)),
V (3)p,q,r(x) = ∂yzUp,q,r(x) = µ2
qµ3r sin(µ
1p(x− a)) cos(µ2
q(y − c)) cos(µ3r(z − e)).
Remark 6.4.1. After obtaining the numerical solution φ(x, t) on the bounded com-
putational domain D, if it is needed to recover the original wave function ψ(x, t) over
a set of fixed grid points in the Cartesian coordinates x, one can use the standard
Fourier/sine interpolation operators from the discrete numerical solution φ(x, t) to
construct an interpolation continuous function over D [38, 141], which can be used
to compute ψ(x, t) over a set of fixed grid points in the Cartesian coordinates x for
any fixed time t ≥ 0.
Remark 6.4.2. If the potential V (x) in (6.7) is replaced by a time-dependent po-
tential, e.g. V (x, t), the rotating Lagrangian coordinates transformation and the nu-
merical method are still valid provided that we replaceW (x, t) in (6.48) byW (x, t) =
V (A(t)x, t) for x ∈ Rd and t ≥ 0.
6.5 Numerical results
In this section, we first test the accuracy of our numerical method and compare
it with previously known methods for computing the dynamics of rotating BECs.
We then study the dynamics of rotating dipolar BECs, including the center of mass,
angular momentum expectation and condensate widths. In addition, the dynamics
of vortex lattices in rotating dipolar BEC are presented. For simplicity, throughout
this section, we consider the 2D GPE (6.7)-(6.8) with the surface density model and
choose the external potential V (x) according to (6.12).
6.5 Numerical results 140
6.5.1 Comparisons of different methods
To test the performance of our method, we compare it with some representative
methods in the literature for computing the dynamics of rotating BECs, includ-
ing the Crank–Nicolson finite difference (CNFD) method [16,90,116], semi-implicit
finite difference (SIFD) method [16, 90] and time-splitting alternating direction im-
plicit (TSADI) method [24]. Note that these methods solve the GPE in Eulerian
coordinates. Here we will not compare our method with those in [18, 21], where
the polar and cylindrical coordinates are used in 2D and 3D, respectively. Since
different computational domains and meshes are used, it is hard to carry out a fair
comparison.
We consider the 2D GPE (6.7)-(6.8) for a rotating BEC without DDI. The pa-
rameters are chosen as d = 2, Ω = 0.4, γx = γy = 1 and η = 0. The initial condition
of the wave function in (6.11) is taken as
ψ0(x) =1
π1/4e
−(x2+2y2)2 , x = (x, y)T ∈ R2. (6.88)
In our simulations, we choose a bounded computational domain D = [−8π, 8π]2.
Let ψ(t) represent the numerical solution with very fine mesh size hx = hy = h =
π/128 and small time step ∆t = 0.0001, and assume it to be a sufficiently good
representation of the exact solution at time t. Denote ψ(hx,hy,∆t)(t) as the numerical
solution at time t obtained with the mesh size (hx, hy) and time step ∆t. Tables
6.1–6.2 show the spatial and temporal errors of different methods, where the errors
are computed as ‖ψ(t) − ψ(h,h,∆t)(t)‖l2 at time t = 1.5. To calculate the spatial
errors in Table 6.1, we always use a very small time step ∆t = 0.0001 so that
the errors from temporal discretization can be neglected compared to those from
spatial discretization. Similarly, the temporal errors are obtained when fine mesh
size hx = hy = π/128 is used.
From Table 6.1, we see that for the same mesh size and time step, the resulting
errors of TSADI and our methods are much smaller than those for CNFD and SIFD
methods. It also indicates that the CNFD and SIFD methods are second-order
6.5 Numerical results 141
Mesh size π/4 π/8 π/16 π/32 π/64
β = 10.15
CNFD 3.895E-1 1.475E-1 4.093E-2 9.662E-3 1.917E-3
SIFD 3.895E-1 1.475E-1 4.093E-2 9.662E-3 1.917E-3
TSADI 1.001E-1 1.969E-3 5.651E-8 9.391E-13 <1.0E-13
Our method 1.031E-1 1.984E-3 5.609E-8 8.907E-13 <1.0E-13
β = 32.60
CNFD 1.2063 4.395E-1 1.722E-1 4.511E-2 8.947E-3
SIFD 1.2064 4.394E-1 1.722E-1 4.511E-2 8.947E-3
TSADI 3.966E-1 2.744E-2 1.875E-5 1.204E-12 <1.0E-12
Our method 4.244E-1 2.785E-2 1.872E-5 1.146E-12 <1.0E-12
β = 80.25
CNFD 1.4181 1.3308 4.806E-1 1.888E-1 4.046E-2
SIFD 1.4180 1.3308 4.806E-1 1.888E-1 4.046E-2
TSADI 1.2239 1.702E-1 1.244E-3 5.208E-9 <1.0E-10
Our method 1.3150 1.731E-1 1.244E-3 5.235E-9 <1.0E-11
Table 6.1: Spatial discretization errors ‖ψ(t)− ψ(hx,hy,∆t)(t)‖l2 at time t = 1.5.
accurate in space while our method and TSADI are spectral-order accurate. On the
other hand, Table 6.2 shows that CNFD, TSADI and our method are second-order
accurate in time, while SIFD is either linear or slightly faster than linear accurate
in time due to the way that we tested the temporal accuracy [10]. In fact, for SIFD
method, second order of accuracy in both space and time was observed when the
mesh size and time step are decreased simultaneously [16]. In general, the spatial
and temporal errors increase with the nonlinearity coefficient β when the mesh size
and time step are fixed. In addition, for fixed β and time step ∆t, in general, the
errors from our method and TSADI are much smaller than those from CNFD and
SIFD.
Compared to the SIFD and CNFD methods, our method has much higher spatial
and/or temporal accuracy and is much easier to be implemented in practical com-
putation. Thus for a given accuracy, the memory and computational costs of our
6.5 Numerical results 142
Time step 1/40 1/80 1/160 1/320 1/640
β = 10.15
CNFD 1.244E-2 4.730E-3 1.195E-3 2.992E-4 7.460E-5
SIFD 6.740E-1 3.107E-2 1.561E-2 7.915E-3 3.908E-3
TSADI 3.510E-4 8.738E-5 2.182E-5 5.448E-6 1.358E-6
Our method 3.433E-4 8.542E-5 2.133E-5 5.326E-6 1.327E-6
β = 32.60
CNFD 1.171E-1 3.898E-2 1.032E-2 2.601E-3 6.491E-4
SIFD 2.589E-1 1.012E-1 3.657E-2 1.625E-2 8.196E-3
TSADI 1.824E-3 4.510E-4 1.124E-4 2.807E-5 6.994E-6
Our method 1.824E-3 4.481E-4 1.117E-4 2.790E-5 6.949E-6
β = 80.25
CNFD 4.716E-1 2.264E-1 7.571E-2 2.024E-2 5.089E-3
SIFD unstable 3.847E-1 1.790E-1 5.719E-2 1.815E-2
TSADI 6.423E-3 1.564E-3 3.887E-4 9.697E-5 2.416E-5
Our method 6.401E-3 1.559E-3 3.874E-4 9.663E-5 2.407E-5
Table 6.2: Temporal discretization errors ‖ψ(t)− ψ(hx,hy,∆t)(t)‖l2 at time t = 1.5.
method can be significantly reduced from those of the SIFD and CNFD methods.
Similarly, compared to TSADI method, the computational cost per time step of our
method is slightly smaller. In addition, the temporal accuracy of our method can be
much easier improved to higher order, e.g. fourth order or higher, by adopting the
standard higher order time-splitting method [15, 67] since our method decomposed
the GPE into two sub-problems in a first order splitting, while the TSADI has to
decompose the GPE into three sub-problems [18].
6.5.2 Dynamics of center of mass
In the following, we study the dynamics of the center of mass by directly simulat-
ing the GPE (6.7)–(6.8) in 2D with SDM long-range interaction (6.6) and harmonic
potential (6.12). To that end, we take d = 2, β = 30√10/π, η = −15
2and dipole
6.5 Numerical results 143
axis n = (1, 0, 0)T . The initial condition in (6.11) is taken as
φ0(x) = α ζ(x− x0), with ζ(x) = (x+ iy)e−(x2+y2)
2 , x ∈ D, (6.89)
where the constant α is chosen to satisfy the normalization condition ‖ψ0‖2 = 1.
Initially, we take x0 = (1, 1)T . In our simulations, we use the computational domain
D = [−16, 16]2, the mesh size ∆x = ∆y = 1/16 and the time step size τ = 0.0001.
We consider the following two sets of trapping frequencies: (i) γx = γy = 1, and
(ii) γx = 1, γy = 1.1. Fig. 6.3 shows the trajectory of the center of mass xc(t) in
the original coordinates as well as the time evolution of its coordinates for different
angular velocities Ω, where γx = γy = 1. On the other hand, Fig. 6.5.2 presents
the same quantities for γx = 1 and γy = 1.1. In addition, the numerical results
are compared with analytical ones from solving the ODEs in (6.33)–(6.35). Figs.
6.3–6.5.2 show that if the external trap is symmetric, i.e., γx = γy, the center of
mass always moves within a bounded region which is symmetric with respect to the
trap center (0, 0)T . Furthermore, if the angular velocity Ω is rational, the movement
is periodic with a period depending on both the angular velocity and the trapping
frequencies. In contrast, when γx 6= γy, the dynamics of the center of mass become
more complicated. The simulation results in Figs. 6.3–6.5.2 are consistent with
those obtained by solving the ODE system in Lemma 6.2.3 for given Ω, γx, and
γy [159] and those numerical results reported in the literatures by other numerical
methods [18, 21, 24].
On the other hand, we also study the dynamics of the center of mass xc(t) in the
new coordinates. When γx = γy and Ω arbitrary, the center of mass has a periodic
motion on the straight line segment connecting −x0 and x0. This is also true for
xc(t) with Ω = 0 (cf. Fig. 6.3). However, the trajectories are different for different
Ω if γx 6= γy. This observations agree with the results in Lemma 6.2.4.
In addition, our simulations show that the dynamics of the center of mass are
independent of the interaction coefficients β and η, which is consistent with Lemma
6.2.3.
6.5 Numerical results 144
-1 0 1
-1
0
1
xc
y c
Ω = 0
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
t
x c(t) o
r yc(t)
Ω = 0
-1 0 1
-1
0
1
xc
y c
Ω = 0.5
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
t
x c(t) o
r yc(t)
Ω = 0.5
-1 0 1
-1
0
1
xc
y c
Ω = 1
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
t
x c(t) o
r yc(t)
Ω = 1
-1 0 1
-1
0
1
xc
y c
Ω = π
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
t
x c(t) o
r yc(t)
Ω = π
Figure 6.3: Results for γx = γy = 1. Left: trajectory of the center of mass, xc(t) =
(xc(t), yc(t))T for 0 ≤ t ≤ 100. Right: coordinates of the trajectory xc(t) (solid
line: xc(t), dashed line: yc(t)) for different rotation speed Ω, where the solid and
dashed lines are obtained by directly simulating the GPE and ‘*’ and ‘o’ represent
the solutions to the ODEs in Lemma 6.2.3.
6.5 Numerical results 145
-1 0 1
-1
0
1
xc
y c
Ω = 0
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
t
x c(t) o
r yc(t)
Ω = 0
-1 0 1
-1
0
1
xc
y c
Ω = 0.5
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
t
x c(t) o
r yc(t)
Ω = 0.5
-4 -2 0
0
1
xc
y c
Ω = 1
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5Ω = 1
t
x c(t) o
r yc(t)
-1 0 1
-1
0
1
xc
y c
Ω = π
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5Ω = π
t
x c(t) o
r yc(t)
Figure 6.4: Results for γx = 1, γy = 1.1. Left: trajectory of the center of mass,
xc(t) = (xc(t), yc(t))T for 0 ≤ t ≤ 100. Right: coordinates of the trajectory xc(t)
(solid line: xc(t), dashed line: yc(t)) for different rotation speed Ω, where the solid
and dashed lines are obtained by directly simulating the GPE and ‘*’ and ‘o’ repre-
sent the solutions to the ODEs in Lemma 6.2.3.
6.5 Numerical results 146
6.5.3 Dynamics of angular momentum expectation and con-
densate widths
To study the dynamics of the angular momentum expectation and condensate
widths, we adapt the GPE (6.7)–(6.8) in 2D with SDM long-range interaction (6.6)
and harmonic potential (6.12), i.e., we take d = 2 and Ω = 0.7. Similarly, the initial
condition in (6.11) is taken as
ψ0(x) = α ζ(x), x ∈ D, (6.90)
where ζ(x) is defined in (6.89) and α is a constant such that ‖ψ0‖2 = 1. In our
simulations, we consider the following four cases:
(i) γx = γy = 1, β = 25√10/π, η = 0, and n = (1, 0, 0)T ;
(ii) γx = γy = 1, β = 25√10/π, η = −15, and n = (1, 0, 0)T ;
(iii) γx = γy = 1, β = 55√10/π, η = −15, and n = (0, 0, 1)T ;
(iv) γx = 1, γy = 1.1, β = 55√
10/π, η = −15, and n = (0, 0, 1)T .
0 5 10 150.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
t
<L
z>(t
)
Case iCase iiCase iiiCase iv
0 5 10 150.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t
Energy: Case i
Energy: Case ii
Energy: Case iii
Energy: Case iv
Mass: Case i-iv
Figure 6.5: Time evolution of the angular momentum expectation (left) and energy
and mass (right) for Cases (i)-(iv) in section 5.3.
In Fig. 6.5, we present the dynamics of the angular momentum expectation,
energy and mass for each of the above four cases in the interval t ∈ [0, 15]. We see
6.5 Numerical results 147
i)0 5 10 15
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
t
δ
x(t)
δy(t)
δr(t)
ii)0 5 10 15
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
t
δ
x(t)
δy(t)
δr(t)
iii)0 5 10 15
1
2
3
4
5
6
7
t
δ
x(t)
δy(t)
δr(t)
iv)0 5 10 15
1
2
3
4
5
6
7
t
δ
x(t)
δy(t)
δr(t)
Figure 6.6: Time evolution of condensate widths in the Cases (i)–(iv) in section 5.3.
that if the external trap is radially symmetric in 2D, then the angular momentum
expectation is conserved when either there is no dipolar interaction (Case (i)) or the
dipolar axis is parallel to the z-axis (Case (iii)). Otherwise, the angular momentum
expectation is not conserved. The above numerical observations are consistent with
the analytical results obtained in Lemma 6.2.1. In addition, we find that our method
conserves the energy and mass very well during the dynamics (cf. Fig. 6.5 right).
Furthermore, from our additional numerical results not shown here for brevity, we
observed that the angular momentum expectation is conserved in 3D for any initial
data if the external trap is cylindrically symmetric and either there is no dipolar
interaction or the dipolar axis is parallel to the z-axis, which can also be justified
mathematically.
The dynamics of the condensate widths are presented in Fig. 6.6. We find that
δr(t) is periodic as long as the trapping frequencies satisfy γx = γy and the influence
of the dipole axis vanishes, e.g. in the Case (i), which confirms the analytical results
of Lemma 6.2.2. Furthermore, from our additional numerical results not shown here
6.5 Numerical results 148
for brevity, we observed that δr(t) is periodic and δx(t) = δy(t) =12δr(t) if η = 0 for
any initial data or n = (0, 0, 1)T for radially symmetric or central vortex-type initial
data.
t = 0 t = 1.5 t = 3
t = 4 t = 5.5 t = 7
Figure 6.7: Contour plots of the density function |ψ(x, t)|2 for dynamics of a vortex
lattice in a rotating BEC (Case (i)). Domain displayed: (x, y) ∈ [−13, 13]2.
6.5.4 Dynamics of quantized vortex lattices
In the following, we apply our numerical method to study the dynamics of quan-
tized vortex lattices in rotating dipolar BECs. Again, we adapt the GPE (6.7)–(6.8)
in 2D with SDM long-range interaction (6.6) and harmonic potential (6.12), i.e.,
we choose d = 2, β = 1000 and Ω = 0.9. The initial datum in (6.11) is chosen
as a stationary vortex lattice which is computed numerically by using the method
in [157, 158] with the above parameters and γx = γy = 1, η = 0, i.e., no long-range
dipole-dipole interaction initially. Then the dynamics of vortex lattices are studied
in two cases:
6.5 Numerical results 149
(i) perturb the external potential by setting γx = 1.05 and γy = 0.95 at t = 0;
(ii) turn on the dipolar interactions by setting η = −600 and dipolar axis n =
(1, 0, 0)T at time t = 0.
In our simulations, we use D = [−16, 16]2, ∆x = ∆y = 1/16 and τ = 0.0001. Figs.
6.7–6.8 show the contour plots of the density function |ψ(x, t)|2 at different time
steps for Cases (i) and (ii), respectively, where the wave function ψ(x, t) is obtained
from φ(x, t) by using interpolation via sine basis (see Remark 6.4.1). We see that
during the dynamics, the number of vortices is conserved in both cases. The lattices
rotate to form different patterns because of the anisotropic external potential and
dipolar interaction in Cases (i) and (ii), respectively. In addition, the results in
Case (i) are similar to those obtained in [18], where a spectral type method in polar
coordinates was used to simulate the dynamics of vortex lattices.
t = 0 t = 1.5 t = 3.5
t = 5.5 t = 8.5 t = 10.5
Figure 6.8: Contour plots of the density function |ψ(x, t)|2 for dynamics of a vortex
lattice in a rotating dipolar BEC (Case (ii)). Domain displayed: (x, y) ∈ [−10, 10]2.
6.6 Conclusions 150
6.6 Conclusions
We proposed a simple and efficient numerical method to simulate the dynamics
of rotating dipolar Bose–Einstein condensation whose properties are described by
the Gross–Pitaevskii equation with both the angular momentum rotation term and
the long-range dipole-dipole interaction. We eliminated the angular rotation term
from the GPE by a rotating Lagrangian coordinate transformation, which makes
it possible to design a simple and efficient numerical method. Based on the new
mathematical formulation under the rotating Lagrangian coordinates, we designed
a numerical method which combines the time-splitting techniques with Fourier/sine
pseudospectral approximation to simulate the dynamics of rotating dipolar BECs.
The new numerical method is explicit, unconditionally stable, spectral order accu-
rate in space and second order accurate in time, and conserves the mass on the
discretized level. Its temporal accuracy can be easily increased to higher order. The
memory cost is O(MK) in 2D and O(MKL) in 3D, and the computational cost per
time step is O (MK ln(MK)) in 2D and O (MKL ln(MKL)) in 3D. More specif-
ically, the method is very easy to implement via fast Fourier transform (FFT) or
discrete sine transform (DST). We reviewed some conserved quantities and dynami-
cal laws of the GPE for rotating BECs and casted their corresponding formulations
in the new rotating Lagrangian coordinates. Based on them, we numerically ex-
amined the conservation of the angular momentum expectation and studied the
dynamics of condensate widths and center of mass for different angular velocities.
In addition, the dynamics of vortex lattice in rotating dipolar BEC were also inves-
tigated. Numerical studies showed that our method is very effective in simulating
the dynamics of rotating BECs without/with DDI.
Chapter 7
Conclusion remarks and future work
In this thesis, by proposing efficient and accurate numerical methods to solve
GLSE and corresponding RDLs, we conducted an extensive numerical study on
quantized vortex phenomena in GLE, NLSE and CGLE on bounded domain with a
small parameter ε. Moreover, we also investigated analytically and numerically on
the the dynamics of rotating dipolar BEC whose properties are described by the GPE
with both the angular rotation term and the long-range dipole-dipole interaction.
In the first part, we studied quantized vortex dynamics and interaction in the
GLSE. Firstly, steady vortex states of the GLSE are reviewed and an efficient and
accurate numerical methods was proposed to simulate GLSE with initial data involv-
ing vortices under different boundary conditions. The numerical method is based
on: (i). applying a time-splitting technique to decouple the nonlinearity in the
GLSE; (ii). for the resulted linear PDE (gradient flow with constant coefficient in
GLE/CGLE case or free Schrodinger equation in NLSE case), if in a rectangular do-
main, for the case of Dirichlet BC, we adapt a fourth-order compact finite difference
method in the spatial discretization and a Crank-Nicolson method in the temporal
discretization, while for the case of Neumann BC, we apply a cosine pseudospectral
method to discretize it; otherwise in a disk domain, we adopt the polar coordinate
in our numerical discretization, then utilized the standard Fourier pseudospectral
discretization in the transverse direction, finite element discretization in the radial
151
152
direction and a Crank-Nicolson discretization in temporal direction. Secondly, we
reviewed various RDLs which govern the motion of the vortex centers to the leading
order, and proposed some methods to solve them. Finally, we applied the proposed
methods to simulate quantized vortex interaction of GLSE with different ε and
initial setups including single vortex, vortex pair, vortex dipole and vortex clusters.
Based on extensive numerical results, we found that the value of ε, the boundary
condition, the geometry of the domain, the initial location of the vortices and the
type of the potential affect the motion of the vortices significantly. Generally, the
boundary effect affect the vortex interaction very much, which lead to very different
nonlinear phenomena from those observed in the case of domain being the whole
plane. Moreover, we verified that the dynamics of vortex centers in the GLSE
dynamics converges to that of the reduced dynamics when ε→ 0 before they collide
and/or move out of the domain and/or after the sound wave propagate away from
them. Surely, after either vortices collide with each other or move out of the domain
or when sound wave is being radiating or propagate back toward them, the RDLs
are no longer valid; however, the dynamics and interaction of quantized vortices
are still physically interesting and they can be obtained from our direct numerical
simulations. Moreover, for each fixed ε, there are regimes which at least depend on
the boundary condition and the geometry of the domain, such that the RDLs failed
qualitatively to describe the vortex motion.
For the case of NLSE, vortices behave like point vortices in ideal fluid on bounded
domains, they never move outside domain. Moreover, we found that the radiation of
NLSE dynamics which is carried by oscillating sound waves modifies the motion of
vortices much, especially in the dynamics of vortex cluster, highly co-rotating vortex
pairs and overlapping vortices. And due to the dispersive and radiative nature, the
RDLs which does not take the radiation into account will be invalid even if there
were small perturbations near around the vortex centers initially.
For the case of the GLE and/or CGLE, which are dissipative systems, vortices
move in a quite different way from the case of NLSE, they can exit the domain in
153
some circumstance. Vortices of like (opposite) winding number will undergo repul-
sive (attractive) interaction, they will move outside domain or merge and annihilate
somewhere in the domain or move toward the boundary and finally stop somewhere
near the boundary. Moreover, there are no radiation or sound waves come up during
the dynamics and interaction of the vortices, the RDLs are still valid if there were
small perturbations in the initial setups. We also investigated the patterns of the
steady states of the vortex clusters, we found that: (i). In the case of Dirichlet BC,
the vortices will all move toward the boundary to form a boundary layer, whose
width is proportional to the value of ε and inverse proportional to the number of
vortices in the cluster. The alignment of the vortices in the steady state depend
on the initial location of the vortices, the boundary condition and the geometry of
the domain. (ii). In the case of Neumann BC, the most of the vortices will exit
the domain, and finally at most one vortex will leftover sitting at the center of the
domain for some proper initial data. However, this steady states is not stable. If we
imposed a slightly perturbation, the vortices will begin to move and finally exit the
domain. These findings confirm the analytical results very well [83, 101]. Further-
more, in the presence of inhomogeneous potential, vortices generally move toward
the critical points of the external potential, and finally stop steady near around
those points, which illustrate clearly the pinning effect.
In the second part, we studied the dynamics of GPE with angular momentum
rotation term and/or the long-range dipolar-dipolar interaction term. Firstly, we
review the two-dimensional (2D) GPE obtained from the 3D GPE via dimension
reduction under anisotropic external potential and derive some dynamical laws re-
lated to the 2D and 3D GPE. By introducing a rotating Lagrangian coordinate
system, the original GPEs are re-formulated to GPEs without the angular momen-
tum rotation which is replaced by a time-dependent potential in the new coordinate
system. We then cast the conserved quantities and dynamical laws in the new rotat-
ing Lagrangian coordinates. Based on the new formulation of the GPE for rotating
BECs in the rotating Lagrangian coordinates, we proposed a time-splitting spectral
154
method for computing the dynamics of rotating BECs. The new numerical method
is explicit, simple to implement, unconditionally stable and very efficient in compu-
tation. It is spectral order accurate in space and second-order accurate in time, and
conserves the mass in the discrete level. Extensive numerical results are reported to
demonstrate the efficiency and accuracy of the new numerical method. Finally, the
numerical method is applied to test the dynamical laws of rotating BECs such as the
dynamics of condensate width, angular momentum expectation and center-of-mass,
and to investigate numerically the dynamics and interaction of quantized vortex
lattices in rotating BECs without/with the long-range dipole-dipole interaction.
The topics that considered here is merely a small part of the world of vortex
dynamics. Many interesting and difficult problems still remains open. The study of
vortex dynamics can always be divided into two groups: (i). either to derive possible
RDLs which are sets of simple ODEs that govern the motion of vortices, (ii). or
to simulate the PDEs directly by some efficient and accurate numerical methods.
Firstly, most of the RDLs reported in the literatures now can be valid only up to the
first collision time and they were derived under some assumptions not so general.
The problem that how to relax those assumptions and extend exist or derive new
RDLs to describe the motion of vortices involving vortex collision, vortex of multiple
degree (and thus splittings and reconnections might happen), vortex exiting domain
and vortices radiating sound waves still remains as a difficult and thus interesting
open problem. This is the first possible direction that we will consider in the future.
Secondly, although the direct simulation can provide us as much information as we
want including vortex splittings, collisions, reconnections and possible sound wave
propagation in dispersive system as well as pinning effect of the vortices, the design
of effective numerical algorithm itself is a difficult issue. The efficiency of the method
presented in this thesis for simulating the GLSE depend on the value of ε, and it
becomes useless for extreme small ε. To propose effective numerical methods which
are ε-independent is another future work.
Moreover, the ideas proposed in this thesis to simulate dipolar rotating BEC is
155
quite simple but effective. We believe that it can be easily extended to study two
component rotating dipolar or spinor BEC [91], etc.
Last but not least, as mentioned in the beginning of this thesis, the GLE we
consider here is the simplified model for model superconductivity, which might not
be that physical interesting, which motivate us to extend our methods to study
the vortex dynamics in full Ginzburg-Landau model that involving electromagnetic
field [133,142]. Other topics such as vortex dynamics in the nonlinear Klein-Gorden
equation [154], the nonlinear Maxwell-Klein-Gorden equation [155], the Landu-
Lifshitz-Gilbert equation [96] and the coupled Ginzburg-Landau equation for mod-
eling unconventional superconductor [105] will also be considered.
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List of Publications
[1] Numerical methods and comparisions for computing dark and bright solitons
in nonlinear Schrodinger equation (with Weizhu Bao and Zhiguo Xu), Journal
of Computational Physics, Vol. 235, pp. 423–445, 2013.
[2] Numerical study of quantized vortex dynamics and interaction in Ginzburg–
Landau equation on bounded domains (with Weizhu Bao), Communication in
Computational Physics, Vol. 14, pp. 819–850, 2013.
[3] Quantized vortex dynamics and interaction in complex Ginzburg–Landau equa-
tion on bounded domains (with Wei Jiang), Applied Mathematics and Com-
putation, Vol. 222, pp. 210-230, 2013.
[4] A simple and efficient numerical method for computing dynamics of rotating
dipolar Bose–Einstein condensation via a rotating Lagrange coordinate (with
Weizhu Bao, Daniel Marahrens and Yanzhi Zhang), SIAM Journal on Scientific
Computing, accepted.
[5] A simple numerical method for computing dynamics of rotating two–component
Bose–Einstein condensation via a rotating Lagrange coordinate (with Ming Ju
and Yanzhi Zhang), Journal of Computational Physics, accepted.
173
List of Publications 174
[6] A Variational–difference numerical method for designing progressive–addition
lenses (with Weizhu Bao, Wei Jiang and Hanquan Wang), Computer–Aided
Design, accepted.
[7] Numerical study of quantized vortex dynamics and interaction in nonlinear
Schrodinger equation on bounded domains (with Weizhu Bao), submitted to
Multiscale Modeling and Simulation: a SIAM Interdisciplinary Journal.